HYDERABAD Centres: Saifabad, Kukatapally, Dilsukhnagar, Narayanaguda, Madhapur, Miyapur - 1 - Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942 NUMBER SYSTEM The Set of Natural Numbers Numbers, which are used in counting are called natural numbers or positive integers. The set of natural numbers are denoted by . 1,2,3,4, ......... The Set of Whole Numbers The set of natural numbers along with number ‘0’ (zero) is called whole numbers. The set of whole numbers are denoted by W . 0,1,2,3, ...... W The Set of Integers The set of natural numbers, their negatives and along with zero are considered as integers denoted by . ....., 3, 2, 1,0,1,2,3, ..... The Set of Rational Numbers The numbers of the form p q , where p and q are integers and 0 q form the set of rational numbers denoted by / , , 0 p pq q q Examples: 2 5 17 0 , , , 5 9 63 etc. Note: The rational numbers are either terminating or recurring decimals Ex: 21 5.25 4 is terminating decimal 40 13.333......, 3 recurring decimal The Set of Irrational Numbers There are numbers, which cannot be expressed in the form p q , where , pq are integers. They are called irrational numbers and are denoted by ' Q or c Q Examples: 1. 2.323323332333…….. 2. 1234.5678910111213…… 3. 2, 3, , e etc The Set of Real Numbers The set of rational numbers together with the set of irrational numbers is called the set of real numbers denoted by .
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NUMBER SYSTEM The Set of Natural Numbersgoing+to+First...are composite numbers 3. 1 is neither prime nor composite Coprime Numbers Two natural numbers are said to be coprime if they
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A real number line is an ordinary geometric straight line whose points have been identified with set of real number.
2 3 01 1 2 2
Here to every real number there corresponds a point on the line and conversely, for every point on the line there corresponds a real number.
Even And Odd Number
Every integer which is exactly divisible by ‘2’ is called even number otherwise it is called odd number.
Every even number is of the form 2n , n is an integer and odd number is of the form 2 1n , n is an integer.
Prime And Composite Numbers
A natural number which has exactly two factors (1 and itself) is called prime number. A natural number which has three or more factors is called a composite number.
Ex: 1. 2,3,5,7,11,13,....... are prime numbers
2. 4,6,8,9,12,15,..... are composite numbers
3. 1 is neither prime nor composite
Coprime Numbers
Two natural numbers are said to be coprime if they don’t have any common factor expect 1.
Ex: 8, 15 are coprime
Intervals
If we wish to consider all the real numbers between a and b (with or without including one or both the points a and b ) then such sets are called intervals.
Open interval (a, b): If a and b are real numbers with a b , we denote by ,a b the set of all real
numbers between a and b excluding a and b
, / ,a b x x a x b
Closed interval [a, b]: If a and b are real numbers with a b , we denote by ,a b the set of all real
numbers between a and b including a and b
, / ,a b x x a x b
Half open intervals: The interval ( , ]a b , which is open at ' 'a and closed at ' 'b is called half open interval
and denotes all real numbers such that a x b similarly [ , )a b denotes all real numbers x such that
A well defined collection of objects is called a set.
Examples
1. The roots of the equation 2 5 6 0x x
2. The English alphabets from a to z
3. The set of natural numbers
In all these collections, we can identify each object precisely and hence they represent sets.
On the other hand, honest people, clever students, handsome boys and so on are relative terms and it is not possible to identify a particular person belongs to the set or not. Hence, they do not form sets in the language of mathematics.
Elements of a Set
The objects that belong to a set are called elements or members of the set.
Set notations
The sets are usually denoted by capital letters , , ,......A B C and their elements by , , ,.......a b c
If a particular element x belong to set A . Then we write it as x A , if x and y belongs to set
A , we write it as ,x y A . However, if an element x does not belong to B , we write x B .
We use curly brackets to enclose the elements of a set. For example
C{set of all even natural numbers} 2,4,6,8,.......
= { /x x is even natural number]
The symbol / stands for ‘such that’
Each element in a set is separated by comma
Specifying sets
(i) Roster method or Listing method
In this method, a set is represented by listing all the elements within . For example
, , , ,A a e i o u
(ii) Set builder method or Rule method
In this method, we state one or more properties of the elements so that we can decide whether an object belongs or not belongs to the set. For example
B { |x x is single digit natural number}
Singleton set
The set which contains only one element is called singleton set. For example
D{ /x x is root of 2 4 4 0x x }
Empty Set or Null Set
The Set which contains no element is called the empty set. The null set is denoted by or for
example E { /x x is real root of 2 1 0x } is the null set.
The cardinal number of a set
If a set A contain finite number of elements n , we denote the cardinal number of set by n A .
In other words, n A stands for number of elements in a finite set A .
For example , , , ,A a e i o u then 5n A
B { /x x is single digit natural number} then 9n B
The cardinal number of the empty set is zero.
Subset of a set
Two sets A and B are such that, each element of set A is also, an element of set B then A is called a subset of B , denoted by A B (read it as A is subset of B)
Thus a set A is a subset of B if x A x B
If A is not a subset of B , we write it as A B
Note: For every set, the empty set and the set itself are subsets.
Example:
For 1,2,3A , the sets , 1 , 2 , 3 , 1,2 , 2,3 , 1,3 , 1,2,3 are all subsets.
Equality of sets
Two sets A and B are said to be equal if every element of set A is an element of set B and every element of set B is an element of set A
i.e x A x B
Note: From the above definition we can conclude the following
1. A set does not change if we change the order in which the elements are tabulated
2. A set does not change if one or more elements are repeated.
For example consider
1,2,3,4 , 3,1,4,2,1,3A B then by definition A B
Proper Subset
Consider the set A which is subset of set B . If there is atleast one element of B which is not in the set A , then A is called proper subset of B and it is denoted by A B
Set of Sets
A set whose elements are set(s) is called set of sets.
Example:
, 1 , 2 , 2,3 is set of sets.
Power Set
Let S is any set. Then the family of the all subsets of S is called the power set of S and is denoted
by P S
Let 1,2,3S , then , 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1,2,3P S
Note: If a set S has ' 'n elements then it power set P S has 2n elements
Universal Set
In any discussion about sets, all the sets under consideration are subsets of a particular set. Such set is called universal set denoted by U .
Algebra of Sets: Let A, B and C be any three sets and ‘U’ be the universal set. Then
Idempotent Laws: a) A A A , b) A A A
Identity Laws: a) A A , b) A U A
Commutative Laws: a) A B B A , b) A B B A
Associative Laws: a) A B C A B C , b) A B C A B C
Distributive Laws: a) A B C A B A C , b) A B C A B A C
De-Morgan Laws: a) ' 'A B A B , b) ' ' 'A B A B
Useful Results:
(i) 'A B A B (ii) 'B A B A (iii) A B A A B
(iv) A B B A B (v) A B B (vi) ' 'A B B A
(vii) A B C A B A C (viii) A B C A B A C
(ix) A B C A B A C (x) A B C A B A C
Venn Diagrams
Venn diagrams are used to represent sets by circles (or any closed shape) inside a rectangle. The rectangle represent the universal set and the circle represent the respective set mentioned. The element of the set are points inside the circle. The Venn diagrams for some sets are as follows.
A B
C
A B C A B C U A
Some important results on number of elements in Sets:
If A, B and C are finite sets and U be the universal sets, then
iv) 2n A B n A n B n A B = Number of elements which belongs to exactly one of A or B
v) n A B C n A n B n C n A B n B C n C A n A B C
vi) Number of elements in exactly two of the sets , , 3A B C n A B n B C n C A n A B C
vii) Number of elements is exactly one of the sets , ,A B C
2 2 2 3n A n B n C n A B n B C n C A n A B C
viii) ' ' 'n A B n A B n U n A B
ix) ' ' 'n A B n A B n U n A B
x) 1 2 ...... nn A A A
i i jn A n A A 11 2....... 1 ....
ni j k nn A A A n A A A
OBJECTIVE QUESTIONS
1. If A and B are two sets containing 3 and 6 elements respectively, then the minimum number of elements in A B is
(A) 3 (B) 4 (C) 6 (D) 12
2. If A and B are two sets containing 4 and 13 elements respectively, then the maximum number of elements in A B is
(A) 13 (B) 17 (C) 4 (D) 21
3. A survey shows that 63% of the Americans like football where as 76% like Basket ball. If %x like both foot ball and basket ball, then
(A) 63x only (B) 39x only (C) 39,40,....,63x (D) none
4. If 2 2 :N x x N and 3 3 :N x x N then 2 3N N =
(A) N (B) 6N (C) 2N (D) 3N
5. If 4 3 1,nx n n N and 9 1 ,y n n N where N is the set of natural numbers, then
X Y is
(A) N (B) X (C) Y (D) Y X
6. If :a ax x and b c d , where ,b c are relatively prime, then
(A) d bc (B) c bd (C) b cd (D) none of these
7. Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of second set. The values of m and n are
9. Let sets X and Y are sets of all positive divisors of 400 and 1000 respectively. Then n X Y
(A) 6 (B) 8 (C) 10 (D) 12
10. If X and Y are two sets, then 'x x y equals
(A) x (B) y (C) (D) none of these
11. If , 1A a , then P A
(A) ,a b (B) , , , ,a b a b (C) , , ,a b A (D) none of these
12. In a group of athletic teams in a school, 21 are in the basket ball team, 26 in the hockey team, and 29 in food ball team. If the play hockey and basket ball; 12 play foot ball and basket ball; 15 play hockey and foot ball and 8 play all the three games, then the total number of players are
(A) 43 (B) 45 (C) 47 (D) 49
13. The number of elements in the set P P P is
(A) 0 (B) 2 (C) 3 (D) 4
14. Let /S x x is an integer between 1 and 1000 (including both) which are neither perfect square
nor perfect cube } Then the number of elements is S are
(A) 959 (B) 960 (C) 961 (D) 962
15. In a town of 10,000 families, it was found that 40% families buy news paper A, 20% families buy newspaper B and 10% families buy newspaper C, 5% buy A and B, 3% buy B and C, and 4% buy C and A. If 2% buy all the three, then the number of families which buy none of A, B and C is
(A) 4,000 (B) 3,300 (C) 4,200 (D) 5,000
16. The average of all the numbers in the set , 1,2,3,.....9S a b a b is
(A) (B) (C) (D)
17. Suppose 1 2 30, ,.....,A A A are 30 sets each having 5 elements and 1 2, ,...., nB B B are n sets each with
3 elements. Let 30
1 1
n
i j
i j
A B S
and each element of S belongs to exactly 10 of the iA ’s and
exactly 9 of the iB ’s . Then n is equal to
(A) 15 (B) 30 (C) 45 (D) none of these
18. Let 1,2,3,4S . The total number of unordered pairs of disjoint subsets of S is equal to
(A) 26 (B) 34 (C) 41 (D) 42
19. If A and B are two sets then CA A B is equal to
(A) (B) A (C) B (D) CA B
20. In a class of 80 students numbered 1 to 80; all odd numbered students opt for cricket, students whose numbers are divisible by 5 opt for football and these whose numbers are divisible by 7 opt for Hockey. The number of students who do not opt any of the three games.