 # Subset | Proper Subset | Superset | Powerset

## 4. Subset

• A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$ .
• It is denoted by
$A \subset B$ if whenever $a \in A$ , then $a \in B$
• If $A \subset B$ and $B \subset A$ , then $A = B$.
• Every set is subset of itself $A \subset A$
• Empty set is subset of every set $\phi \subset A$
• if $A \subset B$ and $B \subset C$ ,then $A \subset C$
Example
1. A ={1,2,3,4,5}
B= {1,2,3,4,5,6,7,8}
Now we can see all the elements of A are in set B
So , $A \subset B$

2. A ={a,b,c}
B= {a,b,c}
Now we can see all the elements of A are in set B
So , $A \subset B$
Now we also see that all the elements of B are in set A
$B \subset A$

3. A ={a,b,c}
B= {1,2,3}
Now we can see none of the element A are present in B
So $A \not\subseteq B$

## 5. Proper Subset and Superset

• If $A \subset B$ and $A \ne B$ , then $A$ is proper subset of $B$.
• We may use symbol $\subseteq$ to define general subset and $\subset$ to define proper subset
• In case of proper subset, $B$ is called superset of set $A$
• it is represented as $B \supset A$
Example
1. A ={1,3,7}
B= {1,3,5,7,8}
Now we can see all the elements of A are in set B and $A \ne B$
So A is proper subset of B
$A \subset B$
and B is the superset of A
$B \supset A$

2. A = ${x : x^2 -1=0}$
B= {1}
Now here A can be represented in set builder form as
A= {-1,1}
We can see all the elements belongs to elements in A
So B is proper subset of A
$B \subset A$
and A is the superset of B
$A \supset B$

## 5. Subset of set of the real numbers

N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers
$T = \left\{ {x:x \in R{\rm{ \; and \;}}x \notin Q} \right\}$, i.e., all real numbers that are not rational
$N \subset Z \subset Q,{\rm{ }}Q \subset R,{\rm{ }}T \subset R,{\rm{ }}N \not\subset T$

## 6. Interval as subset of R Real Number

$a,b \in R,b > a$
 $(a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval $\{ y:a < y < b\}$ $[a,b]$ It is the closed interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, b $\{ x:a \le x \le b\}$ $[a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, but not b $\{ x:a \le x < b\}$ $(a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including b, but not a $\{ x:a < x \le b\}$

## Power Set

• The collection of all subsets of a set $X$ is called the power set of $X$. It is denoted by $P(X)$. In $P(X)$, every element is a set.
• if $X = \left\{ {1,2,3} \right\}$, then
$X\left( A \right) = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {1,3} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\}} \right\}$
Also, note that $n\left[ {X\left( A \right)} \right] = 8 = {2^3}$
• In general, if $X$ is a set with $n\left( X \right) = m$, then it can be shown that
$n\left[ {P\left( A \right)} \right] = {2^m}$
Example
List the subsets of {a,0,-a}
Solution
A={1,0, 1 }.
-The subset of A having no element is the empty set φ
- The subsets of A having one element are { –a }, { 0 }, { a }.
-The subsets of A having two elements are {–a, 0}, {–a, a} ,{0, a}.
-The subset of A having three elements of A is A itself.
So, all the subsets of A are φ, {–a}, {0}, {a}, {–a, 0}, {–a, a},{0, a} and {–a, 0, a}.