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Factors & Multiples
Finding HCF by Division Method Prime and Composite Numbers
An exact divisor of a number is called its factor.
Ex: 1, 2, 3 and 6 are factors of number 6.
The number 1 is a factor of every number.  Every number is a factor of itself.  The factors of a number are either less than or equal to the number itself.  All numbers have a finite number of factors.  The product of two numbers is called a multiple of each of the two numbers being multiplied.  A number is a multiple of all its factors.  Every number is a multiple of 1 and of itself.  There are infinite multiples of a number. If the sum of the factors of a number is two times the number, then the number is called a perfect number.  Numbers that have only two factors in the form of 1 and the number itself are called prime numbers.
Numbers that have more than two factors are called composite numbers.  The number 1 is neither a prime number nor a composite number.  All numbers with 0, 2, 4, 6 or 8 in the unit?s or one?s place are multiples of 2, and are called even numbers.  All numbers with 1, 3, 5, 7 or 9 in the unit?s or one?s place are called odd numbers.
The number 2 is the smallest prime number, and also the only prime number that is even.  All prime numbers, except 2, are odd numbers.   The sum of any two prime numbers, except with 2, is an even number.
Divisibility of Numbers
Factor:An exact divisor of a number is called its factor.
Multiple: The product of two numbers is called a multiple of each of the two numbers being multiplied.
Prime numbers: Numbers with two factors, 1 and itself, are called prime numbers.
Tests of divisibility:
There are certain tests of divisibility that can help us to decide whether a given number is divisible by another number.
Divisibility of numbers by 2:
A number that has 0, 2, 4, 6 or 8 in its ones place is divisible by 2.
Divisibility of numbers by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Divisibility of numbers by 4
A number is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.
Divisibility of numbers by 5
A number that has either 0 or 5 in its ones place is divisible by 5.
Divisibility of numbers by 6:
A number is divisible by 6 if that number is divisible by both 2 and 3.
Divisibility of numbers by 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Divisibility of numbers by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility of numbers by 10:
A number that has 0 in its ones place is divisible by 10.
Divisibility of numbers by 11:
If the difference between the sum of the digits at the odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.
Co?prime numbers:
If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.
General rules of divisibility for all numbers:
Prime Factorisation, HCF and LCM 
Writing a number as a product of its prime factors is called the prime factorisation of the number. Eg: (i)  18=2 x 3 x 3      (ii)  40=2 x 2 x 2 x 5 The greatest of the common factors of the given numbers is called their highest common factor (HCF).  It is also known as the greatest common divisor. Eg: Prime factorisation of 16 = 2 x 2 x 2 x 2 Prime factorisation of 40 = 2 x 2 x 2 x 5 HCF of 16 and 40 = 2 x 2 x 2 = 8 The smallest common multiple of the given numbers is called their Least Common Multiple (LCM). Eg: The LCM of given numbers using their prime factorisation: Prime factorisation of 4 = 2 x 2 Prime factorisation of 6 = 2 x 3 LCM of 4 and 6 = 2 x 2 x 3 =12
To find the LCM of the given numbers using the division method:
Write the given numbers in a row.
Divide the numbers by the smallest prime number that divides one or more of the given numbers.
Write the number that is not divisible, in the second row.
Write the new dividends in the second row.
Divide the new dividends by another smallest prime number.
Continue dividing till the dividends are all prime numbers or 1.Stop the process when all the new dividends are prime numbers or 1. 
Finding HCF by Division Method
Consider the numbers 16 and 24. Factors:16 = 16 × 1    = 8 × 2    = 4 × 4Factors of 16: 16, 8, 4, 2 and 1.24 = 24 × 1    = 12 × 2    = 8 × 3    = 6 × 4Factors of 24: 24, 12, 8, 6, 4, 3, 2 and 1Common factors of 16 and 24 = 8, 4, 2 and 1Largest common factor = 8    ∴ HCF of 16 and 24 = 8The HCF of two or more numbers is a number which is:• A common factor of all the given numbers• The largest among the common factors.HCF is also known as Greatest Common Divisor, that is, GCD.To find the HCF of two or more numbers:• Determine all the factors of each number, and• Identify the largest among the common factors.Methods of finding HCF:• Continued Division method• Common Division methodContinued Division method:i. Divide the larger number by the smaller number.ii. If the remainder is 0, the smaller number, that is the divisor, is the HCF.ii. If the remainder is not 0, divide the divisor by the remainder, which is the new divisor.iv. If the new remainder is 0, the last divisor is the HCF.v. Otherwise repeat the process of dividing the divisor by the remainder till the remainder becomes zero. The last divisor is the HCF.e.g.The numbers 48 and 30.H.C.F of 48 and 30 is 6HCF for three or more numbers:Consider the numbers 135, 150 and 225.i. First, take 135 and 150 and to find HCF of these two numbers.HCF of 135 and 150 = 15ii. Next, take the HCF of the first two numbers obtained, 15 and the remaining number 225.HCF of 15 and 225 = 15HCF of 135, 150 and 225 = 15Consider the numbers 16, 24, 36 and 42.iii. Take any two numbers and find their HCF, say 16 and 24.iv. Take the HCF of the first two numbers obtained and a number from the remaining numbers, and find their HCF.v. Repeat the process till all the given numbers have been considered.The last HCF is the HCF of all the given numbers.NOTE: The HCF of the given numbers will be the same, irrespective of the order in which the numbers are taken for finding HCF.Common Division method:i. Find a prime factor that is common to all the given numbers.ii. Divide all the given numbers with the common prime factor.iii. Identify a common prime factor of the quotients.iv. Divide all the quotients with the common prime factor.v. Repeat this process until the quotients have no common prime factor.The product of the common prime factors is the HCF of all the given numbers.e.g.Consider the numbers 171, 189 and 243.HCF of 171, 189 and 243 = 3 x 3= 9Co-prime NumbersWhen the HCF of two natural numbers is 1, the numbers are called co-prime numbers.Consider the numbers 33 and 49.HCF of 49 and 33 = 1 49 and 33 are co-prime numbers.A few other examples of co-prime numbers arei. 4 and 9.ii. 5 and 12.
Relation Between LCM and HCF 
The LCM of two or more numbers is the smallest natural number that is a multiple of each of the given numbers. The HCF of two or more numbers is the greatest number that divides each one of them exactly. Relation between the HCF and LCM of any two given numbers: Product of LCM and HCF of two given numbers is equal to the product of the numbers.
If p and q are two numbers, p × q = LCM of p and q × HCF of p and q. Formulas obtained by rearranging the equation: LCM of p and q = p × q HCF of p and q HCF of p and q = p × q LCM of p and q
HCF of fractions: The HCF of fractions is defined as (the HCF of the numerators) divided by (the LCM of the denominators). HCF of fractions = HCF of numerators LCM of denominators LCM of fractions: LCM of fractions is defined as (the LCM of the numerators) divided by (the HCF of the denominators). LCM of fractions = LCM of numerators HCF of denominators
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Decimals
Comparing Decimals
If a block of one unit is divided into 10 equal parts, then each part is (one ? tenth) of the unit.  It is written as in decimal representation.  The dot denotes the decimal point. Every fraction whose denominator is 10 can be written in decimal notation.   Eg: If a block of one unit is divided into 100 equal parts, then each part is   of the unit.  It is written as in decimal notation.  Every fraction whose denominator 100 can be written in decimal notation. Eg: To read decimals, we can use the following chart.  The first digit to the right after the decimal point represents the tenths parts, the second the hundredths parts, and so on.
Decimal pointTenthsHundredthsThousandths
.
All decimal numbers can be represented on the number line.  Every decimal number can be represented as a fraction.  Any two decimal numbers can be compared. The comparison starts with the whole part of the numbers. If the whole parts are equal, then the tenth parts can be compared, and so on.  Decimal numbers are used in many ways in real life.  For example, in representing the units of money, length and weight, we use decimal numbers.
Addition and Subtraction of Decimals
To add or subtract decimal numbers, make sure that the decimal points of the given numbers are placed exactly one below another.  While adding or subtracting two decimal numbers, the number of digits after the decimal point should be equal. In case they are not equal, the gaps must be filled with zeros after the last digit. For example:
To add 6.82 and 5
First insert zeros in the empty places after the decimal point so that both the numbers have the same number of digits after the decimal point. Next, write the numbers such that their decimal points are one below another.
To subtract  5 from  6.82
First insert zeros in the empty places after the decimal point so that both the numbers have the same number of digits after the decimal point. Next, write the numbers such that their decimal points are one below another.
Addition or subtraction should be carried out from the extreme right side. Place the decimal point correctly after performing the addition or subtraction.
Multiplication and Division on Decimals
Multiplication of decimals:
To multiply a whole number by a decimal number, follow these steps:
Ignore the decimals and multiply the two numbers.
Count the numbers of digits to the right of decimal point in the original decimal number.
Insert the decimal, from right to left, in the answer by the same count.
Eg: (i)   3×0.2=0.6  (ii)    3×0.4=1.2
To multiply a decimal number by a decimal number, follow these steps:
Ignore the decimals and multiply the two numbers.
Count the number of digits to the right of decimal point in both the decimal numbers.
Add the number of digits counted and insert the decimal, from right to left, in the answer by the same count.
To multiply a decimal number by a decimal number
, follow these steps:Ignore the decimals and multiply the two numbers.Count the number of digits to the right of decimal point in both the decimal numbers.Add the number of digits counted and insert the decimal, from right to left, in the answer by the same count.
To multiply a decimal number by a decimal number, follow these steps:
Ignore the decimals and multiply the two numbers.
Count the number of digits to the right of decimal point in both the decimal numbers.
Add the number of digits counted and insert the decimal, from right to left, in the answer by the same count.
Eg: (i)   0.2×0.7=0.14  (ii)   0.9×0.02=0.018
To multiply a decimal number by 10, 100 or 1000, follow these steps:
While multiplying a decimal number by 10, retain the original number and shift the decimal to the right by one place.
While multiplying a decimal number by 100, retain the original number and shift the decimal to the right by two places.
While multiplying a decimal number by 1000, retain the original number and shift the decimal to the right by three places.
Division of decimals:
To divide a decimal number by a whole number, follow these steps:
Convert the decimal number into a fraction.
Take the reciprocal of the divisor.
Multiply the reciprocal by the fraction.
To divide a decimal number by another decimal number, follow these steps:
Convert both the decimal numbers into fractions.
Take the reciprocal of the divisor.
Multiply the reciprocal by the fraction.
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Fractions
Types of Fractions
A fraction is a part of a whole. A whole can be a group of objects or a single object. For example,   is a fraction.  In this, 3 is called the numerator and 15 is called the denominator.
In the figure shown here, the shaded portion is represented by.   Whole numbers are represented on the number line as shown here:
A fraction can be represented on the number line.   For example,
Consider a fraction is greater than 0, but less than 1.
Divide the space between 0 and 1 into two equal parts. We can show one part as the fraction
Consider another fraction   is greater than 0, but less than 1.
Divide the space between 0 and 1 into five equal parts. We can show the first part as the second as the third as the fourth as and the fifth part as
Proper fractions:
A proper fraction is a number representing a part of a whole.   In a proper fraction, the number in the denominator shows the number of parts into which the whole is divided, while the number in the numerator shows the number of parts that have been taken.   Eg:
Improper fractions:
A fraction in which the numerator is bigger than the denominator is called an improper fraction. Eg:
Mixed fractions:
A combination of a whole and a part is said to be a mixed fraction.
Eg:  
Conversion of improper fraction into mixed fraction:
An improper fraction can be expressed as mixed fraction by dividing the numerator by the denominator of the improper fraction to obtain the quotient and the remainder.  Then the mixed fraction will be.
Conversion of mixed fraction into improper fraction:
A mixed fraction can be written in the form an improper fraction by writing it in the following way:
Like fractions:
Fractions with the same denominator are said to be like fractions.
Eg:
Unlike fractions:
Fractions with different denominators are called unlike fractions.
Eg:
Equivalent fractions:
Fractions that represent the same part of a whole are said to be equivalent fractions.
Eg:
To find an equivalent fraction of a given fraction, multiply both the numerator and the denominator of the given fraction by the same number.
Simplest form of fraction:
A fraction is said to be in its simplest form or its lowest form if its numerator and denominator have no common factor except one.  The simplest form of a given fraction can also be found by dividing its numerator and denominator by its highest common factor (HCF).
Comparing Fractions
Fractions with the same denominator are called like fractions.
Comparing like fractions:
In like fractions, the fraction with the greater numerator is greater.
Two fractions are unlike fractions if they have different denominators.
Comparing unlike fractions:
If two fractions with the same numerator but different denominators are to be compared, then the fraction with the smaller denominator is the greater of the two.
To compare unlike fractions, we first convert them into equivalent fractions.  For example, to compare the following fractions ie., We find the common multiple of the denominators 6 and 8.  48 is a common multiple of 6 and 8. 24 is also a common multiple of 6 and 8.  Least Common Multiple (LCM) of 6 and 8 = 24 x = x =  ⇐   Hence, we can say that is greater than
Addition and Subtraction of Fractions
Like fractions:
If two fractions have the same number in the denominator, then they are said to be like fractions.
To add like fractions:
Add the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the like fractions as the denominator of the resultant fraction.
To subtract like fractions:
Subtract the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the like fractions as the denominator of the resultant fraction.
Unlike fractions:
Fractions with different numbers in the denominators are said to be unlike fractions.
To add unlike fractions:
Find their equivalent fractions with the same denominator.
Add the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the obtained like fraction as the denominator of the resultant fraction.
To subtract unlike fractions:
Find their equivalent fractions with the same denominator.
Subtract the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the obtained like fraction as the denominator of the resultant fraction.
Addition or subtraction of mixed fractions:
Two mixed fractions can be added or subtracted by adding or subtracting the whole numbers of the two fractions, and then adding or subtracting the fractional parts together.  Two mixed fractions can also be converted into improper fractions and then added or subtracted.
Multiplication and Division of Fraction
Multiplication of fractions:
To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction. Eg: While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction. Eg: To multiply two fractions, multiply their numerators and denominators. Eg: When two proper fractions are multiplied, the product is less than each of the individual fractions. When two improper fractions are multiplied, the product is greater than each of the individual fractions.
Division of fractions:
To obtain the reciprocal of a fraction, interchange the numerator with the denominator. Eg: . To divide a whole number by a fraction, take the reciprocal of the fraction and then multiply it by the whole number.
Eg: To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Eg: To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.
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Whole Numbers
The numbers used for counting are called natural numbers.  The number that comes immediately before another number in counting is called its predecessor.  The number that comes immediately after another number in counting is called its successor.  To find the successor of any given natural number, just add 1 to the given number.  The value of nothing is represented by the number zero.  
 Eg: 3 ? 3 = 0Natural numbers together with the number zero are called whole numbers.  When comparing two whole numbers, the number that lies to the right on the number line is greater.  When comparing two whole numbers, the smaller number lies to the left on the number line.
Integers
Whole numbers are represented on the number line as shown here:
If you move towards the right from the zero mark on the number line, the value of thenumbers increases.   If you move towards the left from the zero mark on the number line, the value of the numbers decreases.   The collection of the numbers, that is, ? -3, -2, -1, 0, 1, 2, 3, ?., is called integers. When we need to use numbers with a negative sign, we need to go to the left of zero on the number line.   These numbers are called negative numbers. Examples where these negative numbers are used are temperature scale, water level in a lake or river, level of oil tank, debit account and outstanding dues.   The numbers -1, -2, -3, -4? which are called negative numbers, are also called negative integers.   The number 1, 2, 3, 4 ?s, which are called positive numbers, are also called positive integers.   If we stand at the zero mark on the number line, we can either go left towards negative integers or right towards positive integers.   When we move left towards zero on the number line, the value of positive integers decreases.  When we move left further away from zero on the number line, the value of negative integers decreases.
Addition of integers:
When two positive integers are added, then we get an integer with a positive sign.
Example: (+8) + (+6)= + 14
When two negative integers are added, then we get an integer with a negative sign.
Example: (-3) + (-5) = -8
When a positive integer is added to a negative integer, then we subtract them and put the sign of the greater integer.   The greater integer can be decided by ignoring the signs of the integers. Example: (+4) + (-9) = -5; (+8) + (-3) = 5
Subtraction of integers:
When we subtract a larger positive integer from a smaller positive integer, the difference is a negative integer.
Eg: (+5)-(+8) = -3
To subtract a negative integer from any given integer, we just add the additive inverse of the negative integer to the given integer.
Eg:  (-5)-(-8) = +3 Thus, the subtraction of an integer is the same as the addition of its additive inverse.  Both addition and subtraction of integers can be shown on a number line.
Properties of Integers
Closure property:
Closure property under addition:
Closure property under subtraction:
Closure property under multiplication:
Closure property under division:
Commutative property:
Commutative property under addition:
Commutative property under subtraction:
Commutative property under multiplication:
Commutative property under division:
Associative property:
Associative property under addition:
Associative property under subtraction:
Associative property under multiplication:
Associative property under division:
Distributive property:
Distributive property of multiplication over addition:
Distributive property of multiplication over subtraction:
Identity under addition:
Identity under multiplication:
Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer.   Eg: 3+4=7;-9+7=2
Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer. Eg: -21-(-9)=-12;8-3=5
Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer. Eg: 5×6=30; -9×-3=27
Integers are NOT closed under division, i.e. for any two integers, Eg:
           Addition is commutative for integers.  For any two integers, a and b, a+b=b+a Eg:5+(-6)=5-6=-1; (-6)+5=-6+5=-1 ?5+(-6)=(-6)+5
Subtraction is NOT commutative for integers.  For any two integers, a-b?b-a Eg:8-(-6)=8+6=14; (-6)-8=-6-8=-14 ?8-(-6)?-6-8
Multiplication is commutative for integers.  For any two integers, a and b, ab=ba Eg:9×(-6)=-(9×6)=-54; (-6)×9=-(6×9)=-54 ?9×(-6)=(-6)×9
Division is NOT commutative for integers.  For any two integers, Eg:3/6=1/2;
Addition is associative for integers.  For any three integers, a, b and c, a+(b+c)=(a+b)+c Eg:5+(-6+4)=5-2=3; (5-6)+4=-1+4=3 ?5+(-6+4)=(5-6)+4
Subtraction is associative for integers.  For any three integers, a-(b-c)?(a-b)-c Eg:5?(6-4)=5-2=3; (5-6)-4=-1-4=-5 ?5?(6-4)?(5-6)-4
Multiplication is associative for integers.  For any three integers, a, b and c,          (a×b)×c=a×(b×c) Eg: [(-3)×(-2))×4]=(6×4)=24 [(-3)×(-2×4) ]=(-3×-8)=24 ?[(-3)×(-2))×4]=[(-3)×(-2×4) ]
Division is NOT associative for integers.
For any three integers, a, b and c, a×(b+c) = a×b+a×c Eg: -2 (4 + 3) = -2 (7)  = -14 -2(4+3)=(-2×4)+(-2×3) =(-8)+(-6) =-14
       For any three integers, a, b and c, a×(b-c)= a×b-a×c Eg: -2 (4- 3) = -2 (1)  = -2 -2(4-3)=(-2×4)-(-2×3) =(-8)-(-6) =-2 The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers.
Integer 0 is the identity under addition.  That is, for an integer a, a+0=0+a=a. Eg: 4+0=0+4=4
The integer 1 is the identity under multiplication.  That is, for an integer a, 1×a=a×1=a Eg: (-4)×1=1×(-4)=-4
Operations on Integers
When two positive integers are added, we get a positive integer.   Eg: 8 + 2 = 10 When two negative integers are added, we get a negative integer. Eg:  -6 + (-3) = -9 When a positive and a negative integer are added, the sign of the sum is always the sign of the bigger number of the two, without considering their signs. Eg:  45 + -25 = 20 and ?45 + 20 = -25 The additive inverse of any integer a is ? a, and the additive inverse of (? a) is a. Eg: Additive inverse of (-12) = - (-12) = 12 Subtraction is the opposite of addition, and, therefore, we add the additive inverse of the integer that is being subtracted, to the other integer.   Eg:  23 ? 43 = 23 + Additive inverse of 43 = 23 + (- 43) = - 20
The product of a positive and a negative integer is a negative integer.  The product of two negative integers is a positive integer. If the number of negative integers in a product is even, then the product is a positive integer. Similarly, if the number of negative integers in a product is odd, then the product is a negative integer.
Division is the inverse operation of multiplication.  The division of a negative integer by a positive integer results in a negative integer.  The division of a positive integer by a negative integer results in a negative integer.  The division of a negative integer by a negative integer results in a positive integer.  For any integer p, p multiplied by zero is equal to zero multiplied by p, which is equal to zero.  For any integer p, p divided by zero is not defined, and zero divided by p is equal to zero, where p is not equal to zero.
Application of BODMAS
A numerical expression is a combination of numbers and arithmetic operators. Finding the value of a numerical expression by performing the operations involved in it is called the simplification of the expression. To simplify an expression, follow an order. This is called BODMAS. B → Bracket O → Of D → Division M → Multiplication A → Addition S → Subtraction Bracket First, all brackets are eliminated by simplifying the terms inside the bracket. Of Next, the operation “of” is similar to multiplication, that is, it can be performed by multiplying the terms. Division The division operation after performing the ‘of’ operation. Multiplication All the multiplications involved in the expression. Addition Then, perform all the additions. Subtraction And finally, perform all the subtraction. The number obtained by performing the arithmetic operation in a numerical expression is called the value of the expression. The BODMAS rule to simplify the given expressions. ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = 2 × 2 + 3 10 - 1 2 × 2 3 ÷ 1 3       (Bracket simplified) = 7 10 - 1 2 × 2 3 × 3 1      (Division simplified) =  7 10 - 1     (Multiplication simplified) =   7 - 10 10            (Substraction simplified)   = -3 10 ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = -3 10 Simplify the second expression
( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 = 3 × 2 - 1 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26         (Bracket simplified) = 5 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 =  5 8 + 5 11 × 11 5 + 15 13 ÷ 5 26    ("of" simplified) =   5 8 + 5 11 × 11 7 + 15 13 × 26 5   (Division simplified) = 5 8 + 5 11 × 11 7 + 3 × 2 =  5 8 + 1 + 6              (Multiplication simplified)     = 5 8 + 7 = 5 + 7 × 8 8        (Addition simplified) =  5 + 56 8 = 61 8 = ( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26    = 61 8 Simplification of numerical expressions Simplify:  16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 16.4 + [9.5 ÷ {3.8 of 0.5}× 4] - 1.4     (Round bracket simplified) = 16.4 + [9.5 ÷ 1.9 × 4] - 1.4     ("of" simplified) = 16.4 + [5 × 4] - 1.4     (Division simplified) = 16.4 + 20 - 1.4     (Multiplication simplified) = 36.4 - 1.4 = 35 ∴ 16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 35.
Brackets
In mathematical expressions, brackets convey that: The operations in the brackets are to be carried out first. The expression in brackets is to be treated as a single number. Different types of brackets are used in Mathematics to get the correct values of expressions. Different types of brackets: Removing the brackets Removing the brackets from an expression means simplifying the expressions within the brackets to get a single number and assigning a sign to it.   Brackets are removed in a certain order. The order of removing the brackets is vinculum, round brackets, curly brackets, square brackets. If there is a positive sign before a bracket, then the bracket is removed without changing the signs of the terms within the brackets. If there is a negative sign before a bracket, then the bracket is removed by changing the signs of the terms within the brackets. If there is no sign between brackets, then the multiplication sign is presumed. Inserting the brackets Brackets are inserted in an expression in a certain way. If there is to be a positive sign before the brackets, then the terms within the brackets are written without any change in their signs. If there is to be a negative sign before the brackets, then the terms within the brackets are written after changing their signs. BODMAS Rule A rule for simplifying expressions is the BODMAS rule. BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. These operations are performed sequentially from left to right. The operator ‘of’ represents multiplication. To perform the 'of' operation, the smallest bracket in the given expression is removed. It should be performed before the other mathematical operations. After performing the ‘of’ operation, the DMAS rule should be applied.
Questions & Answers
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1 .
Find the sum of first 30 terms of an A.P.
Sol: Let a be the first term and d be the common difference of A.P then T 2 = a + d = 2 ----------(1) an...
2 .
solve showing all the process
Sol 1) The given number is 10000.   143) 10000 (69         858       -----        1420...
3 .
WHAT IS A NUMBER??
A number is an arithmetical value that is expressed by a word or symbol or figure, representing a particular quantity and used in counting
4 .
solve
10
5 .
why any no. * 0 becomes 0
yes
Application of BODMAS
A numerical expression is a combination of numbers and arithmetic operators. Finding the value of a numerical expression by performing the operations involved in it is called the simplification of the expression. To simplify an expression, follow an order. This is called BODMAS. B → Bracket O → Of D → Division M → Multiplication A → Addition S → Subtraction Bracket First, all brackets are eliminated by simplifying the terms inside the bracket. Of Next, the operation “of” is similar to multiplication, that is, it can be performed by multiplying the terms. Division The division operation after performing the ‘of’ operation. Multiplication All the multiplications involved in the expression. Addition Then, perform all the additions. Subtraction And finally, perform all the subtraction. The number obtained by performing the arithmetic operation in a numerical expression is called the value of the expression. The BODMAS rule to simplify the given expressions. ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = 2 × 2 + 3 10 - 1 2 × 2 3 ÷ 1 3       (Bracket simplified) = 7 10 - 1 2 × 2 3 × 3 1      (Division simplified) =  7 10 - 1     (Multiplication simplified) =   7 - 10 10            (Substraction simplified)   = -3 10 ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = -3 10 Simplify the second expression
( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 = 3 × 2 - 1 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26         (Bracket simplified) = 5 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 =  5 8 + 5 11 × 11 5 + 15 13 ÷ 5 26    ("of" simplified) =   5 8 + 5 11 × 11 7 + 15 13 × 26 5   (Division simplified) = 5 8 + 5 11 × 11 7 + 3 × 2 =  5 8 + 1 + 6              (Multiplication simplified)     = 5 8 + 7 = 5 + 7 × 8 8        (Addition simplified) =  5 + 56 8 = 61 8 = ( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26    = 61 8 Simplification of numerical expressions Simplify:  16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 16.4 + [9.5 ÷ {3.8 of 0.5}× 4] - 1.4     (Round bracket simplified) = 16.4 + [9.5 ÷ 1.9 × 4] - 1.4     ("of" simplified) = 16.4 + [5 × 4] - 1.4     (Division simplified) = 16.4 + 20 - 1.4     (Multiplication simplified) = 36.4 - 1.4 = 35 ∴ 16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 35.
Brackets
In mathematical expressions, brackets convey that: The operations in the brackets are to be carried out first. The expression in brackets is to be treated as a single number. Different types of brackets are used in Mathematics to get the correct values of expressions. Different types of brackets: Removing the brackets Removing the brackets from an expression means simplifying the expressions within the brackets to get a single number and assigning a sign to it.   Brackets are removed in a certain order. The order of removing the brackets is vinculum, round brackets, curly brackets, square brackets. If there is a positive sign before a bracket, then the bracket is removed without changing the signs of the terms within the brackets. If there is a negative sign before a bracket, then the bracket is removed by changing the signs of the terms within the brackets. If there is no sign between brackets, then the multiplication sign is presumed. Inserting the brackets Brackets are inserted in an expression in a certain way. If there is to be a positive sign before the brackets, then the terms within the brackets are written without any change in their signs. If there is to be a negative sign before the brackets, then the terms within the brackets are written after changing their signs. BODMAS Rule A rule for simplifying expressions is the BODMAS rule. BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. These operations are performed sequentially from left to right. The operator ‘of’ represents multiplication. To perform the 'of' operation, the smallest bracket in the given expression is removed. It should be performed before the other mathematical operations. After performing the ‘of’ operation, the DMAS rule should be applied.
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Operations on Sets
Summary
Union of sets Let P and Q be two sets. P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} The union of sets P and Q is a set that consists of the elements from both the sets, P and Q. P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9} Set P union Q consists of all the elements of sets P and Q. Union: The union of two sets A and B is set C, which consists of all elements that are either in A or in B. Symbolically, the union of sets A and B is represented as A ∪ B = {x: x ∈ A or x ∈ B}. The Venn diagram representation of two sets A and B is as shown.
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The region shaded in green represents the union of the sets, A and B. Both the circles are shaded since set A union B consists of all the elements of sets A and B. Properties exhibited by the union of two sets: (i) A ∪ B = B ∪ A (Commutative law) (ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law) (iii) A ∪ ∅ = A (Law of identity element, ϕ is the identity of ∪) (iv) A ∪ A = A (Idempotent law) (v) U ∪ B = U(Law of U) Intersection of sets Consider the sets P and Q, P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} In sets P and Q, the common element is five. Therefore, the intersection of sets P and Q is 5. P ∩ Q = {5} Intersection: The intersection of two sets A and B is the set of all those elements that belong to both A and B. Symbolically, the intersection of sets A and B is represented as A ∩ B = {x: x ∈ A and x ∈ B}. The Venn diagram representation of intersection of two sets A and B is as shown.
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The region shaded in green represents the common elements belonging to the two sets. Property of the intersection of sets Consider two sets C and D C = {2, 4, 6, 8}, D = {3, 5, 7, 9} There is no common element among the sets, C and D. Therefore, C intersection D is a null set. C ∩ D = Ø The Venn diagram representation of sets C and D is as shown.
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Properties of the intersection of two sets 1. A ∩ B = B ∩ A (Commutative law) 2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law) 3. ∅ ∩ A = ∅ , U ∩ A = A (Law of φ and U) 4. A ∩ A = A (Idempotent law) 5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law) Distributive law can be verified with the help of Venn diagrams: LHS of the equation A ∩ (B ∪ C) can be represented as shown below:
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RHS of the equation:
(A ∩ B) can be represented as:
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(A ∩ C) can be represented as:
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(A ∩ B) ∪ (A ∩ C) can be represented as
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Difference of two sets Consider two sets X and Y. X = {2, 3, 4, 5, 6} Y = {5, 6, 7, 8} Difference of sets X and Y taken in the same order (X - Y), contains the elements that are present in X but not in Y. Difference of sets Y and X taken in the same order (Y - X), contains the elements that are present in Y but not in X. It can be observed that, X - Y ≠ Y - X. Representation of set X - Y through a Venn diagram:
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Only the region of set X, which is not shared by set Y, is shaded. Representation of set Y - X through a Venn diagram:
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Only the region of set Y that is not shared by set X is shaded. The difference of any two sets A and B taken in order is represented symbolically as: A – B = {x: x ∈ A and x ∉ B} Similarly, B minus A is written symbolically as: B – A = {x: x ∈ B and x ∉ A} Complement of a set Complement of a set: Let U be the universal set and A be a subset of U. Then the complement of A is the set of all elements of U that are not the elements of A. The complement of a set A is symbolically represented as: A' = {x: x ∈ U and x ∉ A}. The Venn diagram representation of the complement of a set is as shown.
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The region shaded in green represents the complement of set A. The complement of a set depends on the universal set. Ex: The complement of the set of natural numbers with the universal set being the set of integers. N = {1,2,3,....} Z = {.., -2, -1, 0 , 1, 2,....} N' = {.., -2, -1, 0} Observations on complements of sets: If A is a subset of the universal set U, then its complement A′ is also a subset of U. The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements. DeMorgan’s law: (A ∪ B) = A' ∪ B' and (A ∩ B) = A' ∩ B' Complement laws: (i) A ∪ A' = U (ii) A ∩ A' = ∅ Law of double complementation: (A' )' = A Laws of empty set and universal set:  ∅' = U and U' = ∅.
Application of Sets
Summary
Results based on intersection, union and difference of two sets. 1) n(A ∪ B) = n(A) + n(B) A ∩ B = ∅ Example: Consider two sets A and B. A = {1,2,3,4} B = {5,6,7,8,9} A ∩ B = ∅ n(A) = 4 n(B) = 5 A ∪ B = {1,2,3,4,5,6,7,8,9} n(A ∪ B) = 9 n(A ∪ B) = n(A) + n(B) ⇒ 9 = 4 + 5 = 9
Therefore, the relation is verified. 2) n(A ∪ B) = n(A) + n(B) - n(A ∩ B) A and B are finite sets. Verification of the relation with the help of a Venn diagram: It can be observed that, (A - B) ∩ (A ∩ B) ∩ (B - A) = ∅ (A - B) ∩ (A ∩ B) ∪ (B - A) = A ∪ B n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) + n(A ∩ B) - n(A ∩ B)
n(A ∪ B) = n(A) + n(B) - n(A ∩ B) Hence, the relation is verified. 3) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C) n(A ∪ D) = n(A) + n(D) - n(A ∩ D) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n(A ∩ (B ∪ C)) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n((A ∩ B) ∪ (A ∩ C)) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(B ∩ C) - n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C) .
Venn Diagrams
Summary
A Venn diagram is a visual representation of sets.
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The concept of Venn diagrams was introduced by an English mathematician, John Venn.
Venn diagrams are represented with the help of closed figures such as circles, ellipses and rectangles.
Venn diagram for the sets A = {2, 5, 6, 7}, B = {3, 8, 9, 10}
One of the universal sets of these two sets is the set of natural numbers.
Universal set: N
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Example            A = Set of prime numbers less than 10
           B=Set of prime divisors of 26
A = {2, 3, 5, 7} B = {2, 13}
These sets are represented in circles A and B as shown.
The element common to both the sets is two.
So, the circles overlap on each other
The common region of the two circles will have the element two, which is common to both the circles.
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Venn Diagrams
Summary
A Venn diagram is a visual representation of sets.
Tumblr media
The concept of Venn diagrams was introduced by an English mathematician, John Venn.
Venn diagrams are represented with the help of closed figures such as circles, ellipses and rectangles.
Venn diagram for the sets A = {2, 5, 6, 7}, B = {3, 8, 9, 10}
One of the universal sets of these two sets is the set of natural numbers.
Universal set: N
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Example            A = Set of prime numbers less than 10
           B=Set of prime divisors of 26
A = {2, 3, 5, 7} B = {2, 13}
These sets are represented in circles A and B as shown.
The element common to both the sets is two.
So, the circles overlap on each other
The common region of the two circles will have the element two, which is common to both the circles.
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Decimals
Comparing Decimals
If a block of one unit is divided into 10 equal parts, then each part is (one ? tenth) of the unit.  It is written as in decimal representation.  The dot denotes the decimal point. Every fraction whose denominator is 10 can be written in decimal notation.   Eg: If a block of one unit is divided into 100 equal parts, then each part is   of the unit.  It is written as in decimal notation.  Every fraction whose denominator 100 can be written in decimal notation. Eg: To read decimals, we can use the following chart.  The first digit to the right after the decimal point represents the tenths parts, the second the hundredths parts, and so on.
Decimal pointTenthsHundredthsThousandths
.
All decimal numbers can be represented on the number line.  Every decimal number can be represented as a fraction.  Any two decimal numbers can be compared. The comparison starts with the whole part of the numbers. If the whole parts are equal, then the tenth parts can be compared, and so on.  Decimal numbers are used in many ways in real life.  For example, in representing the units of money, length and weight, we use decimal numbers.
Addition and Subtraction of Decimals 
dd or subtract decimal numbers, make sure that the decimal points of the given numbers are placed exactly one below another.  While adding or subtracting two decimal numbers, the number of digits after the decimal point should be equal. In case they are not equal, the gaps must be filled with zeros after the last digit. For example:
To add 6.82 and 5
First insert zeros in the empty places after the decimal point so that both the numbers have the same number of digits after the decimal point. Next, write the numbers such that their decimal points are one below another.
To subtract  5 from  6.82
First insert zeros in the empty places after the decimal point so that both the numbers have the same number of digits after the decimal point. Next, write the numbers such that their decimal points are one below another.
Addition or subtraction should be carried out from the extreme right side. Place the decimal point correctly after performing the addition or subtraction.
Multiplication and Division on Decimals
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Fractions
Types of Fractions
A fraction is a part of a whole. A whole can be a group of objects or a single object. For example,   is a fraction.  In this, 3 is called the numerator and 15 is called the denominator.
In the figure shown here, the shaded portion is represented by.   Whole numbers are represented on the number line as shown here:
A fraction can be represented on the number line.   For example,
Consider a fraction is greater than 0, but less than 1.
Divide the space between 0 and 1 into two equal parts. We can show one part as the fraction
Consider another fraction   is greater than 0, but less than 1.
Divide the space between 0 and 1 into five equal parts. We can show the first part as the second as the third as the fourth as and the fifth part as
Proper fractions:
A proper fraction is a number representing a part of a whole.   In a proper fraction, the number in the denominator shows the number of parts into which the whole is divided, while the number in the numerator shows the number of parts that have been taken.   Eg:
Improper fractions:
A fraction in which the numerator is bigger than the denominator is called an improper fraction. Eg:
Mixed fractions:
A combination of a whole and a part is said to be a mixed fraction.
Eg:  
Conversion of improper fraction into mixed fraction:
An improper fraction can be expressed as mixed fraction by dividing the numerator by the denominator of the improper fraction to obtain the quotient and the remainder.  Then the mixed fraction will be.
Conversion of mixed fraction into improper fraction:
A mixed fraction can be written in the form an improper fraction by writing it in the following way:
Like fractions:
Fractions with the same denominator are said to be like fractions.
Eg:
Unlike fractions:
Fractions with different denominators are called unlike fractions.
Eg:
Equivalent fractions:
Fractions that represent the same part of a whole are said to be equivalent fractions.
Eg:
To find an equivalent fraction of a given fraction, multiply both the numerator and the denominator of the given fraction by the same number.
Simplest form of fraction:
A fraction is said to be in its simplest form or its lowest form if its numerator and denominator have no common factor except one.  The simplest form of a given fraction can also be found by dividing its numerator and denominator by its highest common factor (HCF).
Comparing Fractions
Fractions with the same denominator are called like fractions. Comparing like fractions:In like fractions, the fraction with the greater numerator is greater.Two fractions are unlike fractions if they have different denominators.Comparing unlike fractions:If two fractions with the same numerator but different denominators are to be compared, then the fraction with the smaller denominator is the greater of the two. To compare unlike fractions, we first convert them into equivalent fractions.  For example, to compare the following fractions ie., We find the common multiple of the denominators 6 and 8.  48 is a common multiple of 6 and 8. 24 is also a common multiple of 6 and 8.  Least Common Multiple (LCM) of 6 and 8 = 24 x  =   x  =    ⇐       Hence, we can say that  is greater than 
Addition and Subtraction of Fractions
Like fractions:
If two fractions have the same number in the denominator, then they are said to be like fractions.
To add like fractions:
Add the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the like fractions as the denominator of the resultant fraction.
To subtract like fractions:
Subtract the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the like fractions as the denominator of the resultant fraction.
Unlike fractions:
Fractions with different numbers in the denominators are said to be unlike fractions.
To add unlike fractions:
Find their equivalent fractions with the same denominator.
Add the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the obtained like fraction as the denominator of the resultant fraction.
To subtract unlike fractions:
Find their equivalent fractions with the same denominator.
Subtract the numerators of the fractions to get the numerator of the resultant fraction.
Use the common denominator of the obtained like fraction as the denominator of the resultant fraction.
Addition or subtraction of mixed fractions:
Two mixed fractions can be added or subtracted by adding or subtracting the whole numbers of the two fractions, and then adding or subtracting the fractional parts together.  Two mixed fractions can also be converted into improper fractions and then added or subtracted.
Multiplication and Division of Fraction
Multiplication of fractions:
To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction. Eg: While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction. Eg: To multiply two fractions, multiply their numerators and denominators. Eg: When two proper fractions are multiplied, the product is less than each of the individual fractions. When two improper fractions are multiplied, the product is greater than each of the individual fractions.
Division of fractions:
To obtain the reciprocal of a fraction, interchange the numerator with the denominator. Eg: . To divide a whole number by a fraction, take the reciprocal of the fraction and then multiply it by the whole number.
Eg: To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Eg: To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.
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Numbers
Closure property:Closure property under addition:Closure property under subtraction:Closure property under multiplication:Closure property under division:Commutative property:Commutative property under addition:Commutative property under subtraction:Commutative property under multiplication:Commutative property under division: Associative property:Associative property under addition: Associative property under subtraction:Associative property under multiplication:Associative property under division:Distributive property:Distributive property of multiplication over addition:Distributive property of multiplication over subtraction:Identity under addition:Identity under multiplication:Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer.   Eg: 3+4=7;-9+7=2Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer. Eg: -21-(-9)=-12;8-3=5Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer. Eg: 5×6=30; -9×-3=27Integers are NOT closed under division, i.e. for any two integers, Eg:             Addition is commutative for integers.  For any two integers, a and b, a+b=b+a Eg:5+(-6)=5-6=-1; (-6)+5=-6+5=-1 ?5+(-6)=(-6)+5Subtraction is NOT commutative for integers.  For any two integers, a-b?b-a Eg:8-(-6)=8+6=14; (-6)-8=-6-8=-14 ?8-(-6)?-6-8Multiplication is commutative for integers.  For any two integers, a and b, ab=ba Eg:9×(-6)=-(9×6)=-54; (-6)×9=-(6×9)=-54 ?9×(-6)=(-6)×9Division is NOT commutative for integers.  For any two integers, Eg:3/6=1/2; Addition is associative for integers.  For any three integers, a, b and c, a+(b+c)=(a+b)+c Eg:5+(-6+4)=5-2=3; (5-6)+4=-1+4=3 ?5+(-6+4)=(5-6)+4Subtraction is associative for integers.  For any three integers, a-(b-c)?(a-b)-c Eg:5?(6-4)=5-2=3; (5-6)-4=-1-4=-5 ?5?(6-4)?(5-6)-4Multiplication is associative for integers.  For any three integers, a, b and c,          (a×b)×c=a×(b×c) Eg: [(-3)×(-2))×4]=(6×4)=24 [(-3)×(-2×4) ]=(-3×-8)=24 ?[(-3)×(-2))×4]=[(-3)×(-2×4) ]Division is NOT associative for integers. For any three integers, a, b and c, a×(b+c) = a×b+a×c Eg: -2 (4 + 3) = -2 (7)  = -14 -2(4+3)=(-2×4)+(-2×3) =(-8)+(-6) =-14        For any three integers, a, b and c, a×(b-c)= a×b-a×c Eg: -2 (4- 3) = -2 (1)  = -2 -2(4-3)=(-2×4)-(-2×3) =(-8)-(-6) =-2 The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers.Integer 0 is the identity under addition.  That is, for an integer a, a+0=0+a=a. Eg: 4+0=0+4=4The integer 1 is the identity under multiplication.  That is, for an integer a, 1×a=a×1=a Eg: (-4)×1=1×(-4)=-4Properties of Integers Whole Numbers
The numbers used for counting are called natural numbers.  The number that comes immediately before another number in counting is called its predecessor.  The number that comes immediately after another number in counting is called its successor.  To find the successor of any given natural number, just add 1 to the given number.  The value of nothing is represented by the number zero.  
Eg: 3 ? 3 = 0 Natural numbers together with the number zero are called whole numbers.  When comparing two whole numbers, the number that lies to the right on the number line is greater.  When comparing two whole numbers, the smaller number lies to the left on the number line.
Integers
Whole numbers are represented on the number line as shown here:
If you move towards the right from the zero mark on the number line, the value of thenumbers increases.   If you move towards the left from the zero mark on the number line, the value of the numbers decreases.   The collection of the numbers, that is, ? -3, -2, -1, 0, 1, 2, 3, ?., is called integers. When we need to use numbers with a negative sign, we need to go to the left of zero on the number line.   These numbers are called negative numbers. Examples where these negative numbers are used are temperature scale, water level in a lake or river, level of oil tank, debit account and outstanding dues.   The numbers -1, -2, -3, -4? which are called negative numbers, are also called negative integers.   The number 1, 2, 3, 4 ?s, which are called positive numbers, are also called positive integers.   If we stand at the zero mark on the number line, we can either go left towards negative integers or right towards positive integers.   When we move left towards zero on the number line, the value of positive integers decreases.  When we move left further away from zero on the number line, the value of negative integers decreases.
Addition of integers:
When two positive integers are added, then we get an integer with a positive sign.
Example: (+8) + (+6)= + 14
When two negative integers are added, then we get an integer with a negative sign.
Example: (-3) + (-5) = -8
When a positive integer is added to a negative integer, then we subtract them and put the sign of the greater integer.   The greater integer can be decided by ignoring the signs of the integers. Example: (+4) + (-9) = -5; (+8) + (-3) = 5
Subtraction of integers:
When we subtract a larger positive integer from a smaller positive integer, the difference is a negative integer.
Eg: (+5)-(+8) = -3
To subtract a negative integer from any given integer, we just add the additive inverse of the negative integer to the given integer.
Eg:  (-5)-(-8) = +3 Thus, the subtraction of an integer is the same as the addition of its additive inverse.  Both addition and subtraction of integers can be shown on a number line.
Properties of Integers 
Closure property:
Closure property under addition:
Closure property under subtraction:
Closure property under multiplication:
Closure property under division:
Commutative property:
Commutative property under addition:
Commutative property under subtraction:
Commutative property under multiplication:
Commutative property under division:
Associative property:
Associative property under addition:
Associative property under subtraction:
Associative property under multiplication:
Associative property under division:
Distributive property:
Distributive property of multiplication over addition:
Distributive property of multiplication over subtraction:
Identity under addition:
Identity under multiplication:
Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer.   Eg: 3+4=7;-9+7=2
Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer. Eg: -21-(-9)=-12;8-3=5
Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer. Eg: 5×6=30; -9×-3=27
Integers are NOT closed under division, i.e. for any two integers, Eg:
           Addition is commutative for integers.  For any two integers, a and b, a+b=b+a Eg:5+(-6)=5-6=-1; (-6)+5=-6+5=-1 ?5+(-6)=(-6)+5
Subtraction is NOT commutative for integers.  For any two integers, a-b?b-a Eg:8-(-6)=8+6=14; (-6)-8=-6-8=-14 ?8-(-6)?-6-8
Multiplication is commutative for integers.  For any two integers, a and b, ab=ba Eg:9×(-6)=-(9×6)=-54; (-6)×9=-(6×9)=-54 ?9×(-6)=(-6)×9
Division is NOT commutative for integers.  For any two integers, Eg:3/6=1/2;
Addition is associative for integers.  For any three integers, a, b and c, a+(b+c)=(a+b)+c Eg:5+(-6+4)=5-2=3; (5-6)+4=-1+4=3 ?5+(-6+4)=(5-6)+4
Subtraction is associative for integers.  For any three integers, a-(b-c)?(a-b)-c Eg:5?(6-4)=5-2=3; (5-6)-4=-1-4=-5 ?5?(6-4)?(5-6)-4
Multiplication is associative for integers.  For any three integers, a, b and c,          (a×b)×c=a×(b×c) Eg: [(-3)×(-2))×4]=(6×4)=24 [(-3)×(-2×4) ]=(-3×-8)=24 ?[(-3)×(-2))×4]=[(-3)×(-2×4) ]
Division is NOT associative for integers.
For any three integers, a, b and c, a×(b+c) = a×b+a×c Eg: -2 (4 + 3) = -2 (7)  = -14 -2(4+3)=(-2×4)+(-2×3) =(-8)+(-6) =-14
       For any three integers, a, b and c, a×(b-c)= a×b-a×c Eg: -2 (4- 3) = -2 (1)  = -2 -2(4-3)=(-2×4)-(-2×3) =(-8)-(-6) =-2 The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers.
Integer 0 is the identity under addition.  That is, for an integer a, a+0=0+a=a. Eg: 4+0=0+4=4
The integer 1 is the identity under multiplication.  That is, for an integer a, 1×a=a×1=a Eg: (-4)×1=1×(-4)=-4
Operations on Integers
When two positive integers are added, we get a positive integer.   Eg: 8 + 2 = 10 When two negative integers are added, we get a negative integer. Eg:  -6 + (-3) = -9 When a positive and a negative integer are added, the sign of the sum is always the sign of the bigger number of the two, without considering their signs. Eg:  45 + -25 = 20 and ?45 + 20 = -25 The additive inverse of any integer a is ? a, and the additive inverse of (? a) is a. Eg: Additive inverse of (-12) = - (-12) = 12 Subtraction is the opposite of addition, and, therefore, we add the additive inverse of the integer that is being subtracted, to the other integer.   Eg:  23 ? 43 = 23 + Additive inverse of 43 = 23 + (- 43) = - 20
The product of a positive and a negative integer is a negative integer.  The product of two negative integers is a positive integer. If the number of negative integers in a product is even, then the product is a positive integer. Similarly, if the number of negative integers in a product is odd, then the product is a negative integer.
Division is the inverse operation of multiplication.  The division of a negative integer by a positive integer results in a negative integer.  The division of a positive integer by a negative integer results in a negative integer.  The division of a negative integer by a negative integer results in a positive integer.  For any integer p, p multiplied by zero is equal to zero multiplied by p, which is equal to zero.  For any integer p, p divided by zero is not defined, and zero divided by p is equal to zero, where p is not equal to zero.
Application of BODMAS
A numerical expression is a combination of numbers and arithmetic operators. Finding the value of a numerical expression by performing the operations involved in it is called the simplification of the expression. To simplify an expression, follow an order. This is called BODMAS. B → Bracket O → Of D → Division M → Multiplication A → Addition S → Subtraction Bracket First, all brackets are eliminated by simplifying the terms inside the bracket. Of Next, the operation “of” is similar to multiplication, that is, it can be performed by multiplying the terms. Division The division operation after performing the ‘of’ operation. Multiplication All the multiplications involved in the expression. Addition Then, perform all the additions. Subtraction And finally, perform all the subtraction. The number obtained by performing the arithmetic operation in a numerical expression is called the value of the expression. The BODMAS rule to simplify the given expressions. ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = 2 × 2 + 3 10 - 1 2 × 2 3 ÷ 1 3       (Bracket simplified) = 7 10 - 1 2 × 2 3 × 3 1      (Division simplified) =  7 10 - 1     (Multiplication simplified) =   7 - 10 10            (Substraction simplified)   = -3 10 ( 2 5 + 3 10 ) - 1 2 × 2 3 ÷ 1 3 = -3 10 Simplify the second expression
( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 = 3 × 2 - 1 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26         (Bracket simplified) = 5 8 + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26 =  5 8 + 5 11 × 11 5 + 15 13 ÷ 5 26    ("of" simplified) =   5 8 + 5 11 × 11 7 + 15 13 × 26 5   (Division simplified) = 5 8 + 5 11 × 11 7 + 3 × 2 =  5 8 + 1 + 6              (Multiplication simplified)     = 5 8 + 7 = 5 + 7 × 8 8        (Addition simplified) =  5 + 56 8 = 61 8 = ( 3 4 - 1 8 ) + 5 11 × 22 35 of 7 2 + 15 13 ÷ 5 26    = 61 8 Simplification of numerical expressions Simplify:  16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 16.4 + [9.5 ÷ {3.8 of 0.5}× 4] - 1.4     (Round bracket simplified) = 16.4 + [9.5 ÷ 1.9 × 4] - 1.4     ("of" simplified) = 16.4 + [5 × 4] - 1.4     (Division simplified) = 16.4 + 20 - 1.4     (Multiplication simplified) = 36.4 - 1.4 = 35 ∴ 16.4 + [9.5 ÷ {(8.05 - 4.25) of 0.5}× 4] - 1.4 = 35.
Brackets
In mathematical expressions, brackets convey that: The operations in the brackets are to be carried out first. The expression in brackets is to be treated as a single number. Different types of brackets are used in Mathematics to get the correct values of expressions. Different types of brackets: Removing the brackets Removing the brackets from an expression means simplifying the expressions within the brackets to get a single number and assigning a sign to it.   Brackets are removed in a certain order. The order of removing the brackets is vinculum, round brackets, curly brackets, square brackets. If there is a positive sign before a bracket, then the bracket is removed without changing the signs of the terms within the brackets. If there is a negative sign before a bracket, then the bracket is removed by changing the signs of the terms within the brackets. If there is no sign between brackets, then the multiplication sign is presumed. Inserting the brackets Brackets are inserted in an expression in a certain way. If there is to be a positive sign before the brackets, then the terms within the brackets are written without any change in their signs. If there is to be a negative sign before the brackets, then the terms within the brackets are written after changing their signs. BODMAS Rule A rule for simplifying expressions is the BODMAS rule. BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. These operations are performed sequentially from left to right. The operator ‘of’ represents multiplication. To perform the 'of' operation, the smallest bracket in the given expression is removed. It should be performed before the other mathematical operations. After performing the ‘of’ operation, the DMAS rule should be applied. In mathematical expressions, brackets convey that: The operations in the brackets are to be carried out first. The expression in brackets is to be treated as a single number. Different types of brackets are used in Mathematics to get the correct values of expressions. Different types of brackets: Removing the brackets Removing the brackets from an expression means simplifying the expressions within the brackets to get a single number and assigning a sign to it.   Brackets are removed in a certain order. The order of removing the brackets is vinculum, round brackets, curly brackets, square brackets. If there is a positive sign before a bracket, then the bracket is removed without changing the signs of the terms within the brackets. If there is a negative sign before a bracket, then the bracket is removed by changing the signs of the terms within the brackets. If there is no sign between brackets, then the multiplication sign is presumed. Inserting the brackets Brackets are inserted in an expression in a certain way. If there is to be a positive sign before the brackets, then the terms within the brackets are written without any change in their signs. If there is to be a negative sign before the brackets, then the terms within the brackets are written after changing their signs. BODMAS Rule A rule for simplifying expressions is the BODMAS rule. BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. These operations are performed sequentially from left to right. The operator ‘of’ represents multiplication. To perform the 'of' operation, the smallest bracket in the given expression is removed. It should be performed before the other mathematical operations. After performing the ‘of’ operation, the DMAS rule should be applied.
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Venn Diagrams
Results based on intersection, union and difference of two sets. 1) n(A ∪ B) = n(A) + n(B) A ∩ B = ∅ Example: Consider two sets A and B. A = {1,2,3,4} B = {5,6,7,8,9} A ∩ B = ∅ n(A) = 4 n(B) = 5 A ∪ B = {1,2,3,4,5,6,7,8,9} n(A ∪ B) = 9 n(A ∪ B) = n(A) + n(B) ⇒ 9 = 4 + 5 = 9Therefore, the relation is verified. 2) n(A ∪ B) = n(A) + n(B) - n(A ∩ B) A and B are finite sets. Verification of the relation with the help of a Venn diagram: It can be observed that, (A - B) ∩ (A ∩ B) ∩ (B - A) = ∅ (A - B) ∩ (A ∩ B) ∪ (B - A) = A ∪ B n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) + n(A ∩ B) - n(A ∩ B)n(A ∪ B) = n(A) + n(B) - n(A ∩ B) Hence, the relation is verified. 3) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C) n(A ∪ D) = n(A) + n(D) - n(A ∩ D) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n(A ∩ (B ∪ C)) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n((A ∩ B) ∪ (A ∩ C)) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(B ∩ C) - n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C) .Application of Sets Union of sets Let P and Q be two sets. P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} The union of sets P and Q is a set that consists of the elements from both the sets, P and Q. P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9} Set P union Q consists of all the elements of sets P and Q. Union: The union of two sets A and B is set C, which consists of all elements that are either in A or in B. Symbolically, the union of sets A and B is represented as A ∪ B = {x: x ∈ A or x ∈ B}. The Venn diagram representation of two sets A and B is as shown. The region shaded in green represents the union of the sets, A and B. Both the circles are shaded since set A union B consists of all the elements of sets A and B. Properties exhibited by the union of two sets: (i) A ∪ B = B ∪ A (Commutative law) (ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law) (iii) A ∪ ∅ = A (Law of identity element, ϕ is the identity of ∪) (iv) A ∪ A = A (Idempotent law) (v) U ∪ B = U(Law of U) Intersection of sets Consider the sets P and Q, P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} In sets P and Q, the common element is five. Therefore, the intersection of sets P and Q is 5. P ∩ Q = {5} Intersection: The intersection of two sets A and B is the set of all those elements that belong to both A and B. Symbolically, the intersection of sets A and B is represented as A ∩ B = {x: x ∈ A and x ∈ B}. The Venn diagram representation of intersection of two sets A and B is as shown. The region shaded in green represents the common elements belonging to the two sets. Property of the intersection of sets Consider two sets C and D C = {2, 4, 6, 8}, D = {3, 5, 7, 9} There is no common element among the sets, C and D. Therefore, C intersection D is a null set. C ∩ D = Ø The Venn diagram representation of sets C and D is as shown. Properties of the intersection of two sets 1. A ∩ B = B ∩ A (Commutative law) 2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law) 3. ∅ ∩ A = ∅ , U ∩ A = A (Law of φ and U) 4. A ∩ A = A (Idempotent law) 5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law) Distributive law can be verified with the help of Venn diagrams: LHS of the equation A ∩ (B ∪ C) can be represented as shown below: RHS of the equation: (A ∩ B) can be represented as: (A ∩ C) can be represented as: (A ∩ B) ∪ (A ∩ C) can be represented as Difference of two sets Consider two sets X and Y. X = {2, 3, 4, 5, 6} Y = {5, 6, 7, 8} Difference of sets X and Y taken in the same order (X - Y), contains the elements that are present in X but not in Y. Difference of sets Y and X taken in the same order (Y - X), contains the elements that are present in Y but not in X. It can be observed that, X - Y ≠ Y - X. Representation of set X - Y through a Venn diagram: Only the region of set X, which is not shared by set Y, is shaded. Representation of set Y - X through a Venn diagram: Only the region of set Y that is not shared by set X is shaded. The difference of any two sets A and B taken in order is represented symbolically as: A – B = {x: x ∈ A and x ∉ B} Similarly, B minus A is written symbolically as: B – A = {x: x ∈ B and x ∉ A} Complement of a set Complement of a set: Let U be the universal set and A be a subset of U. Then the complement of A is the set of all elements of U that are not the elements of A. The complement of a set A is symbolically represented as: A' = {x: x ∈ U and x ∉ A}. The Venn diagram representation of the complement of a set is as shown. The region shaded in green represents the complement of set A. The complement of a set depends on the universal set. Ex: The complement of the set of natural numbers with the universal set being the set of integers. N = {1,2,3,....} Z = {.., -2, -1, 0 , 1, 2,....} N' = {.., -2, -1, 0} Observations on complements of sets: If A is a subset of the universal set U, then its complement A′ is also a subset of U. The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements. DeMorgan’s law: (A ∪ B) = A' ∪ B' and (A ∩ B) = A' ∩ B' Complement laws: (i) A ∪ A' = U (ii) A ∩ A' = ∅ Law of double complementation: (A' )' = A Laws of empty set and universal set:  ∅' = U and U' = ∅. A Venn diagram is a visual representation of sets. The concept of Venn diagrams was introduced by an English mathematician, John Venn. Venn diagrams are represented with the help of closed figures such as circles, ellipses and rectangles. Venn diagram for the sets A = {2, 5, 6, 7}, B = {3, 8, 9, 10} One of the universal sets of these two sets is the set of natural numbers. Universal set: N Example            A = Set of prime numbers less than 10            B=Set of prime divisors of 26 A = {2, 3, 5, 7} B = {2, 13} These sets are represented in circles A and B as shown. The element common to both the sets is two. So, the circles overlap on each other The common region of the two circles will have the element two, which is common to both the circles.
Operations on Sets
Union of sets Let P and Q be two sets. P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} The union of sets P and Q is a set that consists of the elements from both the sets, P and Q. P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9} Set P union Q consists of all the elements of sets P and Q. Union: The union of two sets A and B is set C, which consists of all elements that are either in A or in B. Symbolically, the union of sets A and B is represented as A ∪ B = {x: x ∈ A or x ∈ B}. The Venn diagram representation of two sets A and B is as shown. The region shaded in green represents the union of the sets, A and B. Both the circles are shaded since set A union B consists of all the elements of sets A and B. Properties exhibited by the union of two sets: (i) A ∪ B = B ∪ A (Commutative law) (ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law) (iii) A ∪ ∅ = A (Law of identity element, ϕ is the identity of ∪) (iv) A ∪ A = A (Idempotent law) (v) U ∪ B = U(Law of U) Intersection of sets Consider the sets P and Q, P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9} In sets P and Q, the common element is five. Therefore, the intersection of sets P and Q is 5. P ∩ Q = {5} Intersection: The intersection of two sets A and B is the set of all those elements that belong to both A and B. Symbolically, the intersection of sets A and B is represented as A ∩ B = {x: x ∈ A and x ∈ B}. The Venn diagram representation of intersection of two sets A and B is as shown. The region shaded in green represents the common elements belonging to the two sets. Property of the intersection of sets Consider two sets C and D C = {2, 4, 6, 8}, D = {3, 5, 7, 9} There is no common element among the sets, C and D. Therefore, C intersection D is a null set. C ∩ D = Ø The Venn diagram representation of sets C and D is as shown. Properties of the intersection of two sets 1. A ∩ B = B ∩ A (Commutative law) 2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law) 3. ∅ ∩ A = ∅ , U ∩ A = A (Law of φ and U) 4. A ∩ A = A (Idempotent law) 5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law) Distributive law can be verified with the help of Venn diagrams: LHS of the equation A ∩ (B ∪ C) can be represented as shown below: RHS of the equation: (A ∩ B) can be represented as: (A ∩ C) can be represented as: (A ∩ B) ∪ (A ∩ C) can be represented as Difference of two sets Consider two sets X and Y. X = {2, 3, 4, 5, 6} Y = {5, 6, 7, 8} Difference of sets X and Y taken in the same order (X - Y), contains the elements that are present in X but not in Y. Difference of sets Y and X taken in the same order (Y - X), contains the elements that are present in Y but not in X. It can be observed that, X - Y ≠ Y - X. Representation of set X - Y through a Venn diagram: Only the region of set X, which is not shared by set Y, is shaded. Representation of set Y - X through a Venn diagram: Only the region of set Y that is not shared by set X is shaded. The difference of any two sets A and B taken in order is represented symbolically as: A – B = {x: x ∈ A and x ∉ B} Similarly, B minus A is written symbolically as: B – A = {x: x ∈ B and x ∉ A} Complement of a set Complement of a set: Let U be the universal set and A be a subset of U. Then the complement of A is the set of all elements of U that are not the elements of A. The complement of a set A is symbolically represented as: A' = {x: x ∈ U and x ∉ A}. The Venn diagram representation of the complement of a set is as shown. The region shaded in green represents the complement of set A. The complement of a set depends on the universal set. Ex: The complement of the set of natural numbers with the universal set being the set of integers. N = {1,2,3,....} Z = {.., -2, -1, 0 , 1, 2,....} N' = {.., -2, -1, 0} Observations on complements of sets: If A is a subset of the universal set U, then its complement A′ is also a subset of U. The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements. DeMorgan’s law: (A ∪ B) = A' ∪ B' and (A ∩ B) = A' ∩ B' Complement laws: (i) A ∪ A' = U (ii) A ∩ A' = ∅ Law of double complementation: (A' )' = A Laws of empty set and universal set:  ∅' = U and U' = ∅.
Application of Sets
Results based on intersection, union and difference of two sets. 1) n(A ∪ B) = n(A) + n(B) A ∩ B = ∅ Example: Consider two sets A and B. A = {1,2,3,4} B = {5,6,7,8,9} A ∩ B = ∅ n(A) = 4 n(B) = 5 A ∪ B = {1,2,3,4,5,6,7,8,9} n(A ∪ B) = 9 n(A ∪ B) = n(A) + n(B) ⇒ 9 = 4 + 5 = 9
Therefore, the relation is verified. 2) n(A ∪ B) = n(A) + n(B) - n(A ∩ B) A and B are finite sets. Verification of the relation with the help of a Venn diagram: It can be observed that, (A - B) ∩ (A ∩ B) ∩ (B - A) = ∅ (A - B) ∩ (A ∩ B) ∪ (B - A) = A ∪ B n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A) + n(A ∩ B) - n(A ∩ B)
n(A ∪ B) = n(A) + n(B) - n(A ∩ B) Hence, the relation is verified. 3) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C) n(A ∪ D) = n(A) + n(D) - n(A ∩ D) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n(A ∩ (B ∪ C)) n(A ∪ B ∪ C) = n(A) + n(B ∪ C) - n((A ∩ B) ∪ (A ∩ C)) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(B ∩ C) - n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C) .
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Types of Sets
A set is a well-defined collection of objects.
Consider A = {x: x is a natural number and x ≤ 4}
A={1,2,3,4}
The number of elements in this set is four.
This is a finite number. Sets having a finite number of elements are known as non-empty finite sets.
Non-empty Finite Set: If the number of elements in a set S is a natural number, then S is said to be Non-empty Finite Set.
Consider a set that does not contain any element.
E = {x: x is a real solution of x2 + 1 = 0}
This set consists of elements that are real roots of the equation x square plus one is equal to zero.
x2 + 1 = 0 x2 = -1 Empty Set: A set that does not contain any element is called an Empty Set or Null Set or a Void Set.
E = {} = ∅
An empty set is denoted by the symbol — phi.
A = {1, 2, 3, 4} E = {} = ∅ Infinite Set: A set that is empty or consists of a definite number of elements is called Finite Set, otherwise it is called an Infinite Set. N = {x: x is a natural number} N = {1,2,3,4,...} X = {a: a is solution of a2 - 1 = 0} Y = {b: b is an odd integer and -2 < b < 2} a2 - 1 = 0 a2 = 1 a =  +1 or -1 X = [-1, 1] Y = [-1, 1]
Sets X and Y are equal since both have the same number of elements as well as the same type of elements.
Equal Sets: Two sets A and B are said to be equal if they have exactly the same elements and we write A=B. Otherwise, the sets are said to be unequal and we write A ≠ B. A = {1,2,3,4} and B = {a,b,c,d}
Elements of these two sets are different. So, these sets are not equal.
Types of sets - I
Equal sets Two sets A and B are said to be equal if every element of A is in B, and every element of B is in A. writtten as A = B. For equal sets, the order of the elements is irrelevant. Two sets can be said to be equal irrespective of the order of their elements. The repetition of the elements is irrelevant for the equality of sets. e.g. P = {5,6,7,8} Q = {5,6,7,8,7} Cardinal number The number of distinct elements in a set is called its cardinal number. The cardinal number of set A is denoted by n(A). S = {T,R,I,G,O,N,O,M,E,T,R,Y} n(S) = 9. Equivalent sets Two sets are said to be equivalent if their cardinal numbers are equal. A ↔ B Note: Two sets may be equivalent, but not necessarily equal. P = {1,2,3} Q = {r,s,t} P ↔ Q P ≠ Q n(P) = n(Q) Disjoint sets Two sets with no elements in common are known as disjoint sets. A = {1,2,3} , B = {4,5,6} Sets A and B are disjoint. Overlapping sets Two sets are said to be overlapping sets if they have at least one element in common. P = {1,2,3,4,5} , Q = {4,5,6} P and Q are overlapping sets.
Universal Set
A set P is said to be a subset of set Q if every element of P is also an element of Q. This is symbolically represented as P ⊂ Q, if x ∈ P ⇒ x ∈ Q P is a subset of Q if x is an element of P, it implies that x is also an element of Q. P is not a subset of Q. P ⊄ Q 1) If every element of set P is contained in Q and vice-versa, then sets P and Q are equal.      This can be defined symbolically as P ⊂ Q , Q ⊂ P ⇔ P = Q. 2) Every set is a subset of itself.     This is symbolically written as A ⊂ A . 3) A ⊂ B and A ≠ B     A is a proper subset of B.     B is a superset of A. 4) If any set P contains only one element, then P is called a singleton set. Example: {5},{x} 5) An empty set is a subset of every set. Universal Set Consider S = {1, 2, 3} {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} are the subsets of the set S. The set containing these sets as elements is called the power set of set S and is denoted by P(S). P(S) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}} m[P(S)] = 8 If a set S consists of m number of elements, the number of elements of the power set is equal to 2m. Consider B = {3,6} P(B) = {{3},{6},{3,6}} n(B) = 2 n[P(B)] = 22 = 4 The set of natural numbers is a part of another set, that is, the set of real numbers. In this context, the set of real numbers is the universal set. Similarly, for the set of integers, the universal set is the set of real numbers. The set of natural numbers is a subset of whole numbers. The set of rational numbers is a subset of the set of real numbers. Consider the real numbers in the interval between the numbers two and three. The first interval includes the numbers two and three. This interval consists of all the numbers between the numbers two and three. In set-builder notation, the interval is represented as [2,3] = {x:2 ≤ x ≤ 3}. The second interval is all the numbers within the interval two and three, excluding the number two. This is represented in set-builder notation as (2,3] = {x:2 < x ≤ 3}. The third interval is all the numbers within the interval two and three excluding three. This interval is represented in the set-builder form as [2,3) = {x:2 ≤ x < 3}. The fourth interval is all the numbers within the interval two and three, excluding two and three. This interval is represented in the set-builder form as (2,3) = {x:2 < x < 3}. Some sets of real numbers. The interval zero to infinity with zero included is the set of non-negative real numbers. Set of non-negative real numbers = [0, ∞) Set of negative real numbers = (-∞, 0) Set of real numbers = (-∞, ∞) On generalising, we can conclude that [a, b] = {x: a ≤ x ≤ b, a, b ∈ R} (a, b] = {x: a < x ≤ b, a, b ∈ R} (a, b) = {x: a < x < b, a, b ∈ R} [a, b) = {x: a ≤ x < b, a, b ∈ R} (b-a) is the length of the intervals (a, b), (a, b], [a, b) and [a, b].
Representation of Sets
A set is a well-defined collection of objects. By well-defined collection, we mean that we should be able to decide whether a particular object belongs to a definite collection or not. Example of a well-defined collection is natural numbers less than five. This collection of natural numbers is well defined, as we can definitely decide whether a given particular number belongs to this collection or not. An example of a set that is not well defined is a set that consists of clever students. This is a not a well-defined set because the choice of clever students is not clear, since there is no accurate measure for cleverness. A set is denoted with a capital letter. E.g.: A, Q, S The elements of a set are denoted with small letters. e.g.: x, p, s, etc. The word element is used synonymously with the words object and member. A= {p, q, r} The element p belongs to set A. p ∈ A z ∉ A N : The set of natural numbers Z : The set of integers Q : The set of integers R : The set of real numbers Z+ : The set of positive integers Q+ : The set of positive rational numbers R+ : The set of positive real Numbers There are two ways of representing sets. 1. Roster form 2. Set-builder form Roster form This method of representing sets is also called the tabular form. Ex: Prime numbers less than 10. P = {2, 3, 5, 7} Similarly, a set of natural numbers between 20 and 30 is represented as: A = {21, 22, 23, 24, 25, 26, 27, 28, 29} The order of the elements in a set is unimportant. While representing a set in roster form, the elements are generally not repeated. Only distinct elements are written while representing a set in roster form. Set-builder form The set-builder form is a more concise form of a given set. To write the set in the set-builder form, start with a bracket and write the variable. Then a colon is placed and a property is assigned to this number. Ex 1: {2, 3, 5 and 7} is written in set-builder form as P = {x:x is a prime number, x < 10}. The set is named as P and read as P is the set of all x such that x is a prime number less than 10. Ex 2: Represent the set of natural numbers between 20 and 30 using the set-builder form. Set of natural numbers between 20 and 30 = {21, 22, 23, 24, 25, 26, 27, 28, 29} We name this set A. A = {n:n is a natural number, 20 < n < 30} This is read as: A is the set of all n such that n is a natural number and n is lies between 20 and 30.
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Representation of Sets
A set is a well-defined collection of objects. By well-defined collection, we mean that we should be able to decide whether a particular object belongs to a definite collection or not. Example of a well-defined collection is natural numbers less than five. This collection of natural numbers is well defined, as we can definitely decide whether a given particular number belongs to this collection or not. An example of a set that is not well defined is a set that consists of clever students. This is a not a well-defined set because the choice of clever students is not clear, since there is no accurate measure for cleverness. A set is denoted with a capital letter. E.g.: A, Q, S The elements of a set are denoted with small letters. e.g.: x, p, s, etc. The word element is used synonymously with the words object and member. A= {p, q, r} The element p belongs to set A. p ∈ A z ∉ A N : The set of natural numbers Z : The set of integers Q : The set of integers R : The set of real numbers Z+ : The set of positive integers Q+ : The set of positive rational numbers R+ : The set of positive real Numbers There are two ways of representing sets. 1. Roster form 2. Set-builder form Roster form This method of representing sets is also called the tabular form. Ex: Prime numbers less than 10. P = {2, 3, 5, 7} Similarly, a set of natural numbers between 20 and 30 is represented as: A = {21, 22, 23, 24, 25, 26, 27, 28, 29} The order of the elements in a set is unimportant. While representing a set in roster form, the elements are generally not repeated. Only distinct elements are written while representing a set in roster form. Set-builder form The set-builder form is a more concise form of a given set. To write the set in the set-builder form, start with a bracket and write the variable. Then a colon is placed and a property is assigned to this number. Ex 1: {2, 3, 5 and 7} is written in set-builder form as P = {x:x is a prime number, x < 10}. The set is named as P and read as P is the set of all x such that x is a prime number less than 10. Ex 2: Represent the set of natural numbers between 20 and 30 using the set-builder form. Set of natural numbers between 20 and 30 = {21, 22, 23, 24, 25, 26, 27, 28, 29} We name this set A. A = {n:n is a natural number, 20 < n < 30} This is read as: A is the set of all n such that n is a natural number and n is lies between 20 and 30.
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