## SUBSETS

SUBSET: A set P is said to be a subset of a set Q if each element of P is also an element of Q. If P is subset of Q, we write $\text{P}\subseteq \text{Q}$

PROPER SUBSET: If $\text{P}\subseteq \,\text{Q}$ and $\text{P}\ne \text{Q}$ then P is called a proper subset of Q and we write $\text{P}\,\subset \,\text{Q}\text{.}$

Example: Consider $\text{P}=\left\{ 1,\,\,2,\,\,3 \right\}$ and $\text{Q}=\left\{ 1,\,\,2,\,\,3,\,\,4 \right\}$

Here, $\text{P}\subset \,\text{Q}$ but $\text{Q}\not\subset \text{P}$

Every element of P lies in Q, but every element of Q does not lies in P.

If $\text{A}\subseteq \text{B}$ and $\text{B}\subseteq \text{A}\Leftrightarrow \,\text{A}\,\text{=}\,\text{B}$

SOME IMPORTANT PROPERTIES OF SUBSETS

(1) $\text{N}\,\subset \,\text{Z}\,\subset \,\text{Q}\,\subset \,\text{R}\,\subset \,\text{C}$

Where N = Natural number

Z = Integers

Q = Rational Numbers

R = Real Numbers

C = Complex Numbers

(2) Every set is its own subset. $\text{A}\subseteq \text{A}$

(3) The empty set is subset of every set. $\phi \subset \,\text{A}$

(4) The total number of subsets of a finite set containing n element is ${{2}^{\text{n}}}$

POWER SET: The set of all subsets of a given set A is called the power set of A, denoted by $\text{P}\left( \text{A} \right)$. Example: $\text{A}\,\text{=}\,\left\{ \text{1,2} \right\}$

Subsets of A are: $\phi ,\,\,\left\{ 1 \right\},\,\,\left\{ 2, \right\},\,\,\left\{ 1,2 \right\}$

$P\left( A \right)=\left\{ \phi ,\,\,\left\{ 1 \right\},\,\,\left\{ 2 \right\},\,\,\left\{ 1,2 \right\} \right\}$

UNIVERSAL SET: A set that contains all set in a given context is called the universal set. It is denoted by U.

Let $\text{A=}\left\{ \text{1,}\,\text{2,}\,\text{3} \right\}\text{,}\,\text{B=}\left\{ \text{2,}\,\text{3,}\,\text{4,}\,\text{5} \right\}$ and $C=\left\{ 6,\,7 \right\}$$\left\{ 1,2,3,4,5,6,7 \right\}$ and $C=\left\{ 6,\,7 \right\}$

We can consider U as $\left\{ 1,2,3,4,5,6,7 \right\}$

Here, $\text{A}\subseteq U$

$\begin{array}{l}\text{B}\subseteq \,U\\\text{C}\subseteq U\end{array}$