# Laws In Set Theory

The algebra of sets defines the properties and the set-theoretic operations of union,Â laws in sets theory, Â 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 (âˆª) and intersection (âˆ©) satisfy many identities. Several of these identities or “laws in setÂ 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 âˆª Q= Q âˆª P and P âˆ© Q = Q âˆ© P

• #### Associative Laws:

For three sets P,Q,R-

• #### Distributive Laws:

For any three sets P,Q,R-

• #### Identity Laws:

 P âˆªÎ¦=P (Î¦âžœ null set) P âˆ© U = P (U âžœUniversal set)
• #### Idempotent Laws:

For any finite set A;

 (i) A U A = A (ii) A âˆ© A = A
• #### De-Morganâ€™s Law:

If P and Q are two sets, then

 (P âˆª Q)c = Pc âˆ© Qc (P âˆ© Q)c = Pc âˆª Qc P âˆ©P = P and P âˆªP = 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 âˆª B = B âˆª A

Solution:

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

A âˆª B = {2, 5, 6, 8, 11, 16, 23}

B âˆ© A = {2, 5, 6, 8, 11, 16, 23} = P âˆª 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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