Power set, Universal set - Practice Questions & MCQ

Power set
The collection of all subsets of a set $A$ is called the power set of $A$. It is denoted by $P(A)$.
For Example if $\operatorname{set} A=\{a, b, c\}$, then

$
\mathrm{P}(\mathrm{~A})=\{\varphi,\{\mathrm{a}\},\{\mathrm{b}\},\{\mathrm{c}\},\{\mathrm{a}, \mathrm{~b}\},\{\mathrm{b}, \mathrm{c}\},\{\mathrm{c}, \mathrm{a}\},\{\mathrm{a}, \mathrm{~b}, \mathrm{c}\}\}
$

The power set of any set is non-empty.
Each element of a Power set is a set.
Number of elements in $P(A)=$ Number of subsets of $\operatorname{set} A=2^{\text {Number of elements in set } A}$

Universal set
A set that contains all sets in a given context is called the "Universal Set". The universal set is usually denoted by $U$, and all its subsets are denoted by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers.

If $A$ is a set of all tigers in a jungle, and $B$ is a set of all deers in the jungle, then the universal set can be all the animals of that jungle, as all tigers and all deers are subsets of this set.