1.7 Power Set

Power set: The collection of all subsets of a set \(\mathrm{A}\) is called the power set of \(\mathrm{A}\). It is denoted by \(\mathrm{P}(\mathrm{A})\).

For example, if \(\mathrm{A}=\{1,2\}\), then \(\mathrm{P}(\mathrm{A})=\{\phi,\{1\},\{2\},\{1,2\}\}\)
Also, note that \(n[\mathrm{P}(\mathrm{A})]=4=2^{2}\)

Note1: In general, if \(\mathrm{A}\) is a set with \(n(\mathrm{~A})=m\), then it can be shown that \(n[\mathrm{P}(\mathrm{A})]=2^{m}\).

Note2: We know that \(\phi\) is a subset of every set.

Example 1: Write down all the subsets of the set \(\{1,2\}\).

Solution:

(i) So, \(\phi\) is a subset of \(\{1,2\}\).

(ii) We see that \(\{1\}\) is a subset of \(\{1,2\}\).

(ii) We see that \(\{2\}\) is a subset of \(\{1,2\}\).

(iii) Also, we know that every set is a subset of itself. So, \(\{1,2\}\) is a subset of \(\{1,2\}\).

Thus, the set \(\{1,2\}\) has, in all, four subsets, viz. \(\phi,\{1\},\{2\}\) and \(\{1,2\}\). The set of all these subsets is called the power set of \(\{1,2\}\).