Set Theory. CPT Section D Quantitative Aptitude Chapter 7 Brijeshwar Prasad Gupta

Transcription

1 Set Theory CPT Section D Quantitative Aptitude Chapter 7 Brijeshwar Prasad Gupta

2 Learning Objectives Number system Set Theory Set operations Product of Sets MCQ

3 Number system Natural numbers:- N N = {1,2,3..} 0 N, -3 N, ⅔ N, 2 N Whole numbers:- W W = {0,1,2,3.}

4 Number system Integers:- I or Z I = { ,-2,-1,0,1,2,3..} Positive integers:- I + I + = {1,2,3 } Negative integers:- I - I - = {-1,-2,-3 } Remark:- 0 I +, 0 I - but 0 I

5 Number system Prime numbers :- P P = {x:x is divisible by either one or itself not by any other number except 1} or P = {2,3,5,7,11,13,17.}

6 Number system Rational numbers:- Q Q = {x:x can be expressed in the form of P/q, where p & q I but q 0, p & q are prime to each other} Irrational numbers:- Q Q = {x:x can not be expressed in the form of P/q}

7 Number system Real numbers :- R R = A set of all rational and all irrational numbers are real numbers i.e. R = Q U Q

8 Set Theory A Collection of Well defined objects A set of vowels of English alphabet A set of even numbers less than 100 A set of multiple of 5

9 Set Theory A set of vowels of English alphabet A = {a,e,i,o,u}

10 Set Theory A set of even numbers less than 100 B = {2,4,6 98}

11 Set Theory A set of multiple of 5 C = {5,10,15..}

12 Set Theory Representation of sets :- A,B,C And members are placed only in { } Methods of describing a set Tabular (Roster, Enumeration ) Method Selector ( Builder, Rule ) Method Venn Diagram

13 Set Theory Tabular method A = { a, e, i, o, u} B = {2,4,6,8 98} C = {5,10,15 }

14 Set Theory

15 Set Theory Venn Diagram :- Diagrammatical representation by closed polygon usually by Circle & Rectangle

16 Types of sets Finite & infinite set Singleton set Null or void set Equal set Equivalent set

17 Types of Sets Joint & disjoint set Sub set Family of sets Power set Universal set Cardinal number

18 Finite set A set whose elements are countable A = {p, q, r, s} B = {1,3,5, } C = {x:x = 5n where n N}

19 Infinite set A set whose elements are uncountable. A = {2,4,6,8..} B = {x:x is n odd number} C = {x:x = 2 n where n R}

20 Singleton set A set in which there is only single element. A = {p} B = {x:x is a perfect square where 20<x <30} C = {x:x is neither positive nor negative} D= {x:x is an even prime number }

21 Null or void set

22 Equal set Two sets are said to be equal if they have same elements A = {a, e, i, o, u} B = {a, i, u, o, e} C = {a, e, e, e, i, i, o, u} A = B = C Contd.

23 Equal set:continued P = {x:x is a letter of word march } Q = {x:x is a letter of word charm } P = Q Remark:- Repetation and arrangement of element does not effect equality of sets.

24 Equivalent set Two sets are said to be equivalent if they have same number of elements A = {a, e, i, o, u} B = {1,3,5,7,9,} A Ξ B

25 Joint set If two sets have some common elements than they are joint sets A = {a, e, i, o, u} B = { a, b, c, d, e, f} i.e. A B ϕ

26 Disjoint Set Two sets are disjoint if they have no common element A = { a, e, i, o, u} B = { p, q, r, s} i.e. A B = ф

27 Cardinal number Representation of number of elements in a given set. It is represented by n (A) A = {a, e, i, o, u} n (A) = 5

28 Sub Set

29 Sub Set: Remarks

30 Sub set: Remarks 6. All possible sub sets of a given set contains n elements are 2 n. Number of elements Number of sub sets = = = 8 etc.

31 Sub Set A = {a, b, c} Total subsets are 8 {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}, ϕ

32 Family of Sets A set of sets is family of set A = {{a, b}, {2,4,6}, {p, q, r, s}}

33 Power set A family of set contains all possible subsets of a given set A = {1,3,5} P(A) = {{1},{3},{5},{1,3},{3,5},{5,1},{1,3,5},ϕ}

34 Universal set A set contains all the elements of concerning sets. It is represented by either U or E A = {2,4,6,8} B = {1,3,5,7,9} C = {5,10,15,20} E = {1,2,3,..20}

35 Set Operations Union operation Intersection operation Compliment operation Difference of sets Symmetric difference Product of sets

36 Union Operation Union of two sets is represented by A U B, and is consist of all the elements of A or B or Both (Tabular method) A = {a, e, i, o, u} B = {a, b, c, d, e, f} A U B = {a, b, c, d, e, f, i, o, u}

37 Union operation (Selector method) A = {x:x is an even number} B = {x:x is an odd number} A U B = {x:x is a natural number} P = {x:x is multiple of 5 100} Q = {x:x is multiple of 4 100} P U Q = {x:x N where x is divisible by 4 or 5} i.e. x A U B than x A, or x B contd...

38 Venn Diagram: Union operation

39 Properties of Union operation A U E = E If A B than A U B = B Idempotent law A U A = A

40 Properties of Union operation Commutative law A U B = B U A Associative law A U (B U C) = (A U B) U C Identity law A U ϕ = A contd.

41 Tabular method A = {a,e,i,o,u} B = {a,b,c,d,e,f} C = {p,q,r,s} A B = {a,e} A C = ϕ

42 Intersection operation Intersection of two sets is represented by A B and its common elements of A & B. i.e. any element of A B is an element of A & B both

43 Selector Method A = {x:x, x is divisible by 4} B = {x:x, x is divisible by 5} A B = {x:x, x is divisible by 20}

44 Venn diagram:intersection

45 Properties of Intersection Commutative law A B = B A Associative law A (B C) = (A B) C Identity law A E = A

46 Properties of Intersection Zero prop. A ϕ= ϕ Idempotent law A A = A If A B than A B = A (A B) A and (A B) B

47 Common Property of Union and Intersection Distributive law A U (B C) = (A U B) (A U C) A (B U C) = (A B) U (A C)

48 Compliment operation Remark :- To find compliment knowledge of universal set is compulsary Compliment of a set is represented by A or A C Ā or A or ~A or U-A. And is consist of elements which are not in A

49 Tabular method A = {2,4,6,8} E = {1,2,3..10} A = {1,3,5,7,9,10} (Selector method) X A => X A

50 Venn diagram: Complement Operation

51 Properties of compliment A A = ϕ A U A = E E = ϕ and ϕ = E (A ) = A

52 Properties of compliment A B,=> B A DE-MORGAN S LAW (A U B) = A B (A B) = A U B

53 Difference of sets Difference of two sets is represented by either A B or A~ B and is consist of all the elements of A which are Not in B (Tabular method) A = {a,e,i,o,u} B = {a,b,c,d,e,f} A B = {i,o,u} B A = {b,c,d,f}

54 Difference of sets Selector method Venn Diagram

55 Properties of difference of sets A-B A and B-A B A-B, A B and B-A are mutually disjoint sets DE-MORGAN S LAW A-(B U C) = (A-B) (A-C) A-(B C) = (A-B) U (A-C)

56 Symmetric difference It is Represented by A Δ B and is consist of union of A-B and B-A i.e. A Δ B = (A-B) U (B-A) A = {a,e,i,o,u} B = {a,b,c,d,e,f} A B = {i,o,u} B A = {b,c,d,f} A Δ B = {b,c,d,f,i,o,u}

57 Ordered pair A pair of two elements where first element belongs to first set and second element belongs to second set and is represented by (a, b) where a A and b B. Remark :- (a,b) (b,a)

58 Cartesian product set If A and B are any two set than the set of all ordered pair whose first member belongs to set A and Second member belongs to set B is called the Cartesian product of A and B in that order is denoted by A X B and read as A Cross B

59 Cartesian product set A = {a,b,c} B = {p,q} A X B = {(a,p),(a,q),(b,p),(b,q),(c,p),(c,q)} B X A = {(p,a),(p,b),(p,c),(q,a),(q,b),(q,c)} A X B B X A

60 Partition of set Under partition of set a universal set say U is subdivided into sub sets which are disjoint but make into a union U, we can say

61 Number of elements in a finite set In case of disjoint sets n(aub) = n(a) + n(b) In case of joint set n(aub) = n(a) + n(b) n(a B) (AUBUC) = n(a) + n(b)+n(c) n(a B) n(b C) n(a C) + n(a B C) contd..

62 . MCQ s

63 MCQ.1 1. In a group of 20 children, 8 drink tea but not coffee and 13 like tea. The number of children drinking coffee but not tea is (a) 6 (b) 7 (c) 1 (d) none of these Answer:(B)

64 MCQ.2 2.If A has 32 elements, B has 42 elements and A B has 62 elements, the number of elements in A B is (a) 12 (b) 74 (c) 10 (d) none of these Answer: A

65 MCQ.3 3. Given A = {2, 3}, B = {4, 5}, C = {5, 6} then A (B C) is (a) {(2, 5), (3, 5)} (b) {(5, 2), (5, 3)} (c) {(2, 3), (5, 5)} (d) none of these Answer:A

66 MCQ.4 4.In a class of 60 students, 40 students like Maths, 36 like Science, and 24 like both the subjects. Find the number of students who like (i) Maths only. (ii) Science only (iii) either Maths or Science (iv) neither Maths nor Science.

67 Solution Let M = students who like Maths and S = students who like Science Then n( M) = 40, n(s) = 36 and n (M S ) = 24 Hence, (i) n(m) n(m S) = = 16 = number of students like Maths only. (ii) n( S ) n(m S) = = 12 = number of students like Science only. (iii) n(m S) = n(m) + n(s) n(m S) = = 52 = number of students who like either Maths or Science. ( iv) n(m S)c = 60 n(m S ) = = 8 = number of students who like neither Maths nor Science.

68 MCQ.5 5. A A is equal to (a) ϕ (b) A (c) E (d) none of these Answer:(B)

69 MCQ.6 A A is equal to (a) ϕ (b) A, (c) E, Answer: A (d) none of these

70 MCQ.7 A U A is equal to (a) ϕ (b) A, (c) E, (d) none of these Answer:C

71 MCQ.8 (A B)' is equal to (a) (A B)' (b) A B' (c) A' B' Answer:C (d) none of these

72 MCQ.9 A E is equal to (a) A (b) E (c) ϕ (d) none of these Answer:(A)

73 MCQ.10 If E = {1, 2, 3, 4, 5, 6, 7, 8, 9}, the subset of E satisfying 5 + x > 10 is (a) {5, 6, 7, 8, 9} (b) {6, 7, 8, 9} (c) {7, 8, 9}, (d) none of these Answer:B

74 MCQ Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the total 28% used coffee and tea 32% tea and cocoa and 30% coffee and cocoa. Only 6% did none of these. Find the number having all the three. (A) 360 (B) 280 (C) 160 (D) None Answer:(A)

75 Thank you