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# SUBSETS

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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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