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}

Post Author: E-Maths