Power sets — lesson. Mathematics State Board, Class 9.

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The set of all subsets of a set $A$ is said to be power set $A$. It is written as $P(A)$.

Example:

Consider the set with two elements

A = {1, 2}

By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.

That is,

\emptyset or {}

and

{1, 2}

Now let us write the elements and its combination of subsets.

The singleton subsets are

{1}

and

{2}

Therefore, the subsets of set $A$ are

{}, {1}, {2}, {1, 2}

Thus, the set of power sets of $A$,

P (A) = {{}, {1}, {2}, {1, 2}}

Important!

The number of elements in set $A$ is always less than the number of elements in the power set of $A$. That is

n (A) \leq n [P (A)]

Let $A$ be a set with $m$ elements. That is,

n (A) = m

The number of elements in the power set of $A$ is given by $n [P (A)] = 2^{m}$ .
The number of a proper subset of $A$ is given by $n [P (A)] - 1 = 2^{m} - 1$ .

Example:

Consider the set with three elements

B = {a, b, c}

Now we are going to find the number of elements in the power set of $B$.

Set $B$ has three elements. That is, $m = 3$.

The formula for this is given by

n [P (B)] = 2^{m}

Substitute the value of $m$.

n [P (B)] = 2^{3} = 8

Let us verify by the actual method:

By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.

That is,