# 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

- If
*A*⊆*B*, then*A*∪*B*=*A* *x*∉_{ }*A*∪*B*⇔*x*∉*A*or*x*∉*B*- Associative law: (
*A*∪*B*) ∪*C*=*A*∪ (*B*∪*C*) - Distributive law:
*A*∪ (*B*∪*C*) = (*A*∪*B*) ∪ (*A*∪*C*);*A*∪ (*B*∪*C*) = (*A*∪*B*) ∪ (*A*∪*C*)

- i. (
*A*∪*B*) ∪*A*=*A*and (*A*_{ }∪*B*) ∪*B*=*B*;*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*-

*B*≠

*B*-

*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*= (*A*–*B*) ∪ (*B*–*A*).