Question

Find the power set of each of these sets, where a and b are distinct elements?

(a) \(\left\{ {\bf{a}} \right\}\)

(b) \(\left\{ {{\bf{a,b}}} \right\}\)

(c) \(\left\{ {\phi \left\{ \phi \right\}} \right\}\)

Short answer

(a) \(P\left( {\left\{ a \right\}} \right) = \left\{ {\phi ,\left\{ a \right\}} \right\}\)

(b) \(P\left( {\left\{ {a,b} \right\}} \right) = \left\{ {\phi ,\left\{ a \right\},\left\{ b \right\}\left\{ {a,b} \right\}} \right\}\)

(c) \(P\left( {\left\{ {\phi ,\left\{ \phi \right\}} \right\}} \right) = \left\{ {\phi ,\left\{ \phi \right\},\left\{ {\left\{ \phi \right\}} \right\}\left\{ {\phi ,\left\{ \phi \right\}} \right\}} \right\}\)

Step-by-step solution

Power of a set

Given a set S, the power set of S is the set of all subsets of the set S.

Therefore, the power set of S is denoted by P(S).

To determine the power of given set \(\left\{ {\bf{a}} \right\}\) (a)

The power set of \(\left\{ a \right\}\) contains all subsets of \(\left\{ a \right\}\).

Therefore, the empty set \(\phi \) and the set \(\left\{ a \right\}\)itself are subsets of \(\left\{ a \right\}\).

Thus, power of set\(\left\{ a \right\}\) is \(P\left( {\left\{ a \right\}} \right) = \left\{ {\phi ,\left\{ a \right\}} \right\}\).

To determine the power of given set \(\left\{ {{\bf{a,b}}} \right\}\) (b)

The power set of \(\left\{ {a,b} \right\}\) contains all subsets of \(\left\{ {a,b} \right\}\).

Therefore, the empty set \(\phi \) and the set \(\left\{ {a,b} \right\}\)itself are subsets of \(\left\{ {a,b} \right\}\).

Thus, the power of set \(\left\{ {a,b} \right\}\)is \(P\left( {\left\{ {a,b} \right\}} \right) = \left\{ {\phi ,\left\{ a \right\},\left\{ b \right\}\left\{ {a,b} \right\}} \right\}\).

To determine the power of given set \(\left\{ {\phi ,\left\{ \phi  \right\}} \right\}\) (c)

The power set of \(\left\{ {\phi ,\left\{ \phi \right\}} \right\}\) contains all subsets of \(\left\{ {\phi ,\left\{ \phi \right\}} \right\}\).

Therefore, the empty set \(\phi \) and the set \(\left\{ {\phi ,\left\{ \phi \right\}} \right\}\)itself are subsets of \(\left\{ {\phi ,\left\{ \phi \right\}} \right\}\).

Thus, the power of set \(\left\{ {\phi ,\left\{ \phi \right\}} \right\}\)is \(P\left( {\left\{ {\phi ,\left\{ \phi \right\}} \right\}} \right) = \left\{ {\phi ,\left\{ \phi \right\},\left\{ {\left\{ \phi \right\}} \right\}\left\{ {\phi ,\left\{ \phi \right\}} \right\}} \right\}\).