__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, \(\text{A}\subseteq U\)

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

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