# Subset

Any set A is subset of set B if each element of set A is also a member of set B.if set A is subset of set B then we write A⊆B.

Example : B={1, 2, 3, 4, 5, 6, 7,8} and A={3, 4, 5, 6} then we write A⊆B all the elements of A is present in set B so A is subset of B.

2^{n} number of subset we can create from a given set whose number of element is n.

**Proper subset : ** Any set A is proper subset of set B if each element of set A is member of set B and atleast one element of set B is not member of set A. proper subset is denoted by symbol ⊂ . if set A is proper subset of B then we write A⊂B.

## Superset of a set

Set A is Supper set of set B if each element of set B is a member of set A. superset is denoted by symbol ⊇. if A is superset of set B then we write A⊇B.

Example : A={1, 2, 3, 4, 5, 6, 7,8} and B={3, 4, 5, 6} each element of set B is a member of set A so A is superset of set B and we write A⊇B.

## Interval of subset

Interval of subset is describe as a element in between given sebset member. if we have set A={1, 2, 3, 4, 5, 6, 7,8} and subset B={3, 4, 5, 6} then interval of subset B can be written as [ 3, 6]={x:3≥x≤6} means elements between 3 and 6 and also contains 3 and 6 .There are four type of interval

- Closed interval
- Open interval
- Closed-open interval
- Open-closed interval

**Closed interval: ** if a and b are real number then Closed interval of subset is denoted by [a, b] and defined as [a, b]={x:a≥x≤b}. this closed interval defined set of all real numbers from a to b.

Example : [2, 7]={x:2≥x≤7} this closed interval contains all real number between 2 and 7 and also contain 2 and 7.

**Open interval: **if a and b are real number then Open interval of subset is denoted by ]a, b[ and defined as ]a, b[={x:a>x<b}. this open interval defined set of all real numbers between a and b.

Example : ]2, 7[={x:2≥x≤7} this open interval contains all real number between 2 and 7 excluding 2 and 7.

**Closed-open interval: **if a and b are real number then Closed-open interval of subset is denoted by [a, b[ and defined as [a, b[={x:a≥x<b}. this closed-open interval defined set of all real numbers from a to b but not include b.

Example : [2, 7[={x:2≥x≤7} this closed-open interval contains all real number between 2 and 7 and also contain 2 but not contain 7.

**Open-closed interval: **if a and b are real number then Open-closed interval of subset is denoted by ]a, b] and defined as ]a, b]={x:a>x≤b}. this closed interval defined set of all real numbers from a to b but but not include a.

Example : ]2, 7]={x:2≥x≤7} this open-closed interval contains all real number between 2 and 7 and also contain 7 but not contain 2.

### Power set

Collection of all possible subset of given set is known as power set of given set. power set of given set A is denoted by P(A).

Example if A={1,3,5} then power set of set A is P(A)={{1}, {3}, {5}, {1, 3},{1, 5}, {3, 5}, {1, 3, 5}}

### Universal set

Universal set is a set that contain all distinct member of all given subsets.if you have 3 subsets then universal set contain all distinct member of all 3 set or contain all distinct subsets member and other member that not present in given subset.

Example: Let A={1, 3} B={2, 3} and C={1, 4}
then Universal set can be S={1, 2, 3, 4, 5, 6} or S={1, 2, 3, 4, 5}

Universal set is denoted by symbol S, U or Ω