Union of two sets

The union of two sets A and B is the set of all those elements which are either in A or in B or in both. This set is denoted by A B (read as “A union B”).

Symbolically, A B = {x : x A or x B}.

The union of two sets can be represented by a Venn diagram as shown in Figure.

The shaded region represents A B.

Intersection of two sets

The intersection of two sets A and B is the set of all the elements which are common in A and B. This set is denoted by A B (read as “A intersection B”).

Symbolically, A B = {x : x A and x B}.

The intersection of two sets can be represented by a Venn diagram as shown in Figure. The shaded region represents A B.

Some important results: Algebra of two sets

1. If A B, then A B = A
2. x A B x A or x B
3. Associative law: (A B) C = A (B C)
4. Distributive law:
1. A (B C) = (A B) (A C);
2. A (B C) = (A B) (A C)
5. i. (A B) A = A and (A B) B = B;

ii. (A B) A = A and (A B) B = B

Difference of two sets

The difference of two sets A and B (also called “relative complement” of B in A) is the set of all those elements of A which are not elements of B. It is denoted by A - B.

Symbolically, A - B = {x : x A and x B}.

A - B can be represented by Venn diagram as shown in Figure. The shaded region represents A - B.

Remark: Clearly A - BB - A (as evident from above example). Hence difference of two sets is not commutative.

Symmetric difference of two sets

The symmetric difference of two sets A and B, denoted by A Δ B, is defined as A Δ B = (AB) (BA).