Laws In Set Theory

The algebra of sets defines the properties and the set-theoretic operations of union, laws in sets theory,  intersection, and relations and the complementation of set inclusion and set equality. Laws in set theory is essential for solving certain problems in CAT quantitative aptitude.

The binary operations of a set union (∪) and intersection (∩) satisfy many identities. Several of these identities or “laws in set theory” have well-established names. The different laws in set theory are mentioned below. Before knowing the laws, check types of sets for better understanding.

  • Commutative Laws:

For two sets P and Q,

P ∪ Q= Q ∪ P and P ∩ Q = Q ∩ P

 

  • Associative Laws:

For three sets P,Q,R-

(P ∪ Q) ∪ R = P ∪ (Q ∪ R) and

P ∩ (Q ∩ R) = (P ∩Q) ∩ R

  • Distributive Laws:

For any three sets P,Q,R-

P ∪ (Q ∩ R) = (P ∪Q) ∩(P∪ R)

P ∩(Q ∪R) = (P ∩Q) ∪ (P∩ R)

 

  • Identity Laws:

P ∪Φ=P (Φ➜ null set)

P ∩ U = P (U ➜Universal set)

  • Idempotent Laws:

For any finite set A;

(i) A U A = A

(ii) A ∩ A = A

  • De-Morgan’s Law:

If P and Q are two sets, then

(P ∪ Q)c = Pc ∩ Qc

(P ∩ Q)c = Pc ∪ Qc

P ∩P = P and P ∪P = P

Some solved examples on laws in set theory are given below:

Example 1:

Let A = {11, 8, 6, 5, 2, 23} and B = {5, 2, 16}. Prove that A ∪ B = B ∪ A

Solution:

A = {2, 5, 6, 8, 11, 23}
B = {16, 5, 2}

A ∪ B = {2, 5, 6, 8, 11, 16, 23}

B ∩ A = {2, 5, 6, 8, 11, 16, 23} = P ∪ Q

Example 2:

Let S={1,2,3}S={1,2,3}. Write all the possible partitions of SS.

Solution:

Remember that a partition of SS is a collection of nonempty sets that are disjoint and their union is SS. There are 55 possible partitions for S={1,2,3}S={1,2,3}:

{1},{2},{3}{1},{2},{3};
{1,2},{3}{1,2},{3};
{1,3},{2}{1,3},{2};
{2,3},{1}{2,3},{1};
{1,2,3}{1,2,3}.

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