Points to Remember

Set Language

Points to Remember

• A set is a well defined collection of objects.

• Sets are represented in three forms (i) Descriptive form (ii) Set – builder form (iii) Roster form.

• If every element of A is also an element of B, then A is called a subset of B.

• If A⊆B and A≠B, then A is a proper subset of B.

• The power set of the set A is the set of all the subsets of A and it is denoted by P(A).

• The number of subsets of a set with m elements is 2^m.

• The number of proper subsets of a set with m elements is 2^m -1.

• If A∩B = ∅ then A and B are disjoint sets. If A∩B ≠ ∅ then A and B are overlapping.

• The difference of two sets A and B is the set of all elements in A but not in B.

• The symmetric difference of two sets A and B is the union of A-B and B-A.

• Commutative Property

For any two sets A and B,

A∪B=B∪A ; A∩B=B∩A

• Associative Property

For any three sets A, B and C

A∪(B ∪C)=(A∪B)∪C ; A∩(B ∩C)=(A∩B)∩C

• Distributive Property

For any three sets A, B and C

A∩(B ∪C)=(A∩B)∪(A∩C)   Intersection over union

A∪(B ∩C)=(A∪B)∩(A∪C)   Union over intersection

• De Morgan’s Laws for Set Difference

For any three sets A, B and C

A−(B ∪C)=(A−B)∩(A−C)

A−(B ∩C)=(A−B)∪(A−C)

• De Morgan’s Laws for Complementation

Consider an Universal set and A, B are two subsets, then

(A∪B)′ =A′ ∩ B ′ ; (A∩B)′ =A′∪ B′

• Cardinality of Sets

If A and B are any two sets, then n (A ∪ B ) = n(A) + n(B ) −n(A ∩ B)

If A, B and C are three sets, then

n (A ∪ B ∪C) = n (A) + n (B) + n (C) − n (A ∩ B ) − n (B ∩C) − n (A ∩C ) + n (A ∩ B ∩C)