# Different Types of Sets

- A set having finite number of elements is called a
**finite set**. - A set which is not a finite set is called an
**infinite set**.*A*= set of all points on a particular straight line. Here*A*is an infinite set. - A set having no element is called a null set or an empty set or a void set. It is denoted by
*Ï†*or {}. - A set having single element is called a singleton set, e.g., {
*Ï†*}, {1}, {3}. - Two finite sets
*A*and*B*are said to be equivalent if they have the same cardinal number. Thus sets*A*and*B*are equivalent iff*n*(*A*) =*n*(*B*). - Two sets
*A*and*B*are said to be equal if each element of*A*is an element of*B*and each element of*B*is an element of*A.*

*Notes:*- Two sets
*A*and*B*are equal if*x*âˆˆ*A*â‡’*x*âˆˆ*B*and*x*âˆˆ*B*â‡’*x*âˆˆ*A*. - Equal sets are equivalent sets but equivalent sets may or may not be equal.

**Subsets:***A*set*A*is said to be a subset of a set*B*if each element of*A*is also an element of*B*. If*A*is a subset of set*B*, we write*A*âŠ†*B*. Thus,*A*âŠ†*B*â‡”*x*âˆˆ*A*â‡’*x*âˆˆ*B*.- Every set is its own subset.
- Empty set is a subset of each set.
- Let
*A*and*B*be any two sets, then*A*=*B*â‡”*A*âŠ†*B*and*B*âŠ†*A*. - Let
*A*,*B*,*C*be three sets. If*A*âŠ†*B*and*B*âŠ†*C*, then*A*âŠ†*C*.

- A set
*A*is said to be a**proper subset**of a set*B*if*A*is a subset of*B*and*A*â‰*B*. - A set
*A*is said to be a superset of set*B*, if*B*is a subset of*A,*i.e., each element of*B*is an element of*A*. If*A*is a superset of*B*, we write*A*âŠ‡*B*. - The set or family of all the subsets of a given set
*A*is said to be the**power set**of*A*and is denoted by*P*(*A*). Symbolically,*P*(*A*) = {*X*:*X*âŠ†*A*}. Thus,*X*âˆˆ*P*(*A*) â‡”*X*âŠ†*A*. If*A*has*n*elements then its power set*P*(*A*) has 2elements, i.e.,^{n}*A*âŠ†*B*â‡’*P*(*A*) âŠ†*P*(*B*). - In any discussion in set theory we need a set such that all sets under consideration in that discussion are its subsets. Such a set is called the
**universal set**for that discussion.