Operations on Sets

Union of sets

If two sets are given, a set can be formed by using all the elements of the two sets. Such a collection is called the union of the given sets.

Definition:

Union of two sets A and B is the set of all elements which are in A, or in B, or both in A and B.

In Set builder form,

Examples:

(1) Let A = {3, 4, 5, 6}, B = {6, 7, 8} and C = {8, 9, 7}.

Then

Intersection of sets

A set can be formed by using all the common elements of two given sets. Such a collection is called the intersection of the given sets.

Definition:

Intersection of two sets A and B is a set whose elements belong to both A and B.

Examples:

(1) Let A = {3, 4, 5, 6}, B = {5, 6, 7}, C = {7, 8, 9}

then

Disjoint sets

Example:

Difference of two sets (Relative complement)

The relative complement of set B in set A is the complement of B in A.

Definition:

If A and B are any two sets then the relative complement of B in A is the set of all elements in A which are not in B. It is denoted by A - B.

Alternate Definition:

Difference of two sets A and B, A - B is a set whose elements belong to A but not to B. A - B is called the relative complement of B w.r.t. A.

Example:

Symmetric Difference of two sets

If A and B are two sets, we define their symmetric difference as the set of all elements that belong to A or to B, but not both A and B, and we denote it by A  D B. Thus

Complement of a set

Let U be a universal set, A be any subset of U, then the elements of U which are not in A i.e., U - A is the complement of A w.r.t. U is written as A' = U - A = A^c.

Examples: