De Morgan's Law, aUb, union of sets, intersection of sets,complement of a set

AU(BC)=(AUB)∩(AUC): Union of Sets is distributive over intersection of sets.

A∩(BUC)=(A∩B)U(A∩C): Intersection of sets is distributive over union of sets.

We can make it still easier.

DEMORGAN’S LAWS:

(AUB)^l=A^l∩B^l The complement of union of sets in the intersection of their complements.

(A∩B)^l=A^IUB^I The complement of intersection of sets is the union of their complements.

A={Prime numbers < 12}, i.e. A= {2,3,5,7,11}

B={x/x Є N 2 ≤ 5}, i. e. B = {2,3,4,5}

C={Perfect square number < 16}, i.e. C={1,4,9}

In Disjoint sets: A∩B ={} or ∅ and n (A∩B) = 0

A set is a collection of well defined objects.

· A set can be represented in two ways they are Rule Method and Roster Method.

Ex. Rule Method

A={x/x is a Prime Number < 8}

Roster Method

B={2,3,5,7}

· Types of sets

a) Finite Set: A set which has countable number of elements

b) Infinite Set: A set in which the number of elements cannot be counted

c) Singleton Set: A set which contains only one element

d) Null Set (Empty Set): A set with no element in it

· Properties

Commutative Property: a) AUB=BUA →Union

b) A∩B=B∩A → Intersection

Distributive Property: a) AU(B∩C)=(AUB)∩(AUC)

Union of sets is distributive over intersection of sets

· Demorgan’s Laws: a) (AUB)^I = A^I ∩ B^I

The complement of union of sets is the intersection of their complements

b) (A∩B)^I = A^I ∪ B^I

The complement of intersection of intersection of sets is the union of their complements

· n(A) represents the number of elements in set A.

· n(AUB) = n(A)+n(B)-n(A∩B)

· n(A∩B) = n(A)+n(B)-n(AUB)

Note: 1) If P and Q are disjoint sets: P∩Q = ø, P-Q = P. Q-P = Q

2) If A and B are equal sets, then

AUB = A or B

1. Union of sets is distributive over intersection of sets can be represented as

a) AUB = BUA

b) (AUB)UC = AU(BUC)

c) AU(B∩C) = (AUB)∩(AUC)

d) A∩(BUC) = (A∩B) U (A∩C)

2. If P={1,2,3,4} and Q={3,4,5} then P-Q is

a) {0,1,2} b) {1,6} c) {1,2} d) {3,4}

3. A is a subset of ‘U’. If n(U) = 12 and n(A^I) = 7, then n(A) is

a) 19 b) 12 c) 7 d) 5

4. Out of 1000 students 750 play cricket, 350 play volley ball, 150 play both the games. Number of students who do not play any of these 2 games is

a) 500 b) 250 c) 200 d) 50

5. If A = {a, b, c, d, e}, B={ a, c, e, g, h} and C={c ,f, g} then A ∩ (BUC) =

a) {c} b) {b, d} c) {e, g} d) {a, c, e}

6. If A={1, 2, 3, 4}, B={2, 3, 5} then n(A∩B) =

a) 2 b) 3 c) 4 d) 7

7. U = {2, 3, 5, 6, 10}, A= {5,6} then diagram which represents A^I is

8. If A = {x N/1 ≤ x ≤ 4}, and B= {3, 4, 5} then A ∩ B is

a) {1, 2, 3, 4, 5} b) {1, 2, 3, 4, 5} c) {1, 2} d) {3, 4}

9. If (AUB)^I = {2, 4, 6}, then A^I ∩ B^I is equal to

a) {1, 2, 3, 4, 5, 6} b) {2, 4, 6} c) {1, 3, 5} d) { }

10. With respect to the given Venn Diagram n(A) is

a) 8 b) 5 c) 7 d) 13

A∩B = A or B A – B = ø B – A = ø

3) If B is a proper subset of A [B is a non zero set] then AUB = A, A∩B = B, B - A = ø

4) If B is a null set or empty set and A is an other set, then AUB = A, A∩B = B, A-B=A, B-A=B

· If A and B are disjoint sets [No elements is common]: A∩B = ø, n(A∩B) = 0 and n(AUB) = n(A) +n(B)