Power Set

Definition
of power set:

We have defined a set as a collection of its elements so, if
S is a set then the collection or family of all subsets of S is called the
power set of S and it is denoted by P(S).

Thus, if S = a, b then the power set of S is given by P(S)
= {{a}, {b}, {a, b}, ∅}

We have defined a set as a collection of its elements if the element be sets themselves, then we have a family of set or set of sets.

Thus, A = {{1}, {1, 2, 3}, {2}, {1, 2}} is a family of sets.

The null set or empty set having no element of its own is an element of the power set; since, it is a subset of all sets. The set being a subset of itself is also as an element of the power set.

For example:

1. The collection
of all subsets of a non-empty set S is a set of sets. Thus, the power set of a
given set is always non-empty. This set is said to be the power set of S and is
denoted by P(S). If S contains N elements, then P(S) contains 2^n
subsets, because a subset of P(S) is either ∅ or a subset containing r elements of S,
r = 1, 2, 3, ……..

Let S = {1, 2, 3} then the power set of S is given by P(S) =
{{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, ∅, S}.

2. If S = (a),
then P(S) = {(a), ∅}; if again S = ∅, then P(S) = {∅}.
It should be notated that ∅ ≠ {∅}. If S = {1, 2, 3} then the
subset of S {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅.

Hence, P(S) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},
{1, 2, 3}, ∅}.

3. We know, since
a set formed of all the subset of a set M as its elements is called a power set
of M and is symbolically denoted by P(M). So, if M is a void set ∅,
then P(M) has just one element ∅ then the power set of M is given by P(M) = {∅}