In Mathematics, a set is defined as a collection of well-defined objects. For example, the set of natural numbers between 1 and 10, the set of even numbers less than 20. If we change the order of writing the elements in a set, it does not make any changes in the set. If one or more elements of a set are repeated, then the set remains the same. In this article, we will learn the important properties of sets operations.
Let A and B be two sets, then we can define a set A intersection B denoted by A⋂B, whose elements consist of all the common elements of A and B. Another set A union B denoted by A⋃B, is the set which contains all the elements of A and B. A compliment denoted by AC, is the set of numbers of universal set U, other than the elements of A. Null set is the set which does not have any elements. It is denoted by ∅. The operations of sets satisfy many identities.
What are the Basic Properties of Sets?
Property 1. Commutative property
Intersection and union of sets satisfy the commutative property.
A⋂B = B⋂A
A⋃B = B⋃A
Property 2. Associative property
Intersection and union of sets satisfy the associative property.
(A⋂B)⋂C = A⋂(B⋂C)
(A⋃B)⋃C = A⋃(B⋃C)
Property 3. Distributive property
Intersection and union of sets satisfy the distributive property.
A⋃(B⋂C) = (A⋃B)⋂(A⋃C)
A⋂(B⋃C) = (A⋂B)⋃(A⋂C)
Property 4. Identity
A⋃∅ = A
A⋂U = A
Property 5. Complement
A⋃AC = U
A⋂AC = ∅
Property 6. Idempotent
A⋂A = A
A⋃A = A