For a set \\(S\\), the power set, which is written as \\(P(S)\\) or \\(2^S\\) refers to the collection of all subsets of \\(S\\).

For example, \\(2 ^ {\lbrace 10, 20 \rbrace} = \lbrace

\lbrace \rbrace,

\lbrace 10 \rbrace,

\lbrace 20 \rbrace,

\lbrace 10,20 \rbrace

\rbrace \\).

How big is \\(2^S\\)? Well, if \\(S\\) has \\(n\\) elements, the power set \\(2^S\\) has \\(2^n\\) elements:

\\[|2^S| = 2^{|S|}\\]

Hence the notation.

For example, \\(2 ^ {\lbrace 10, 20 \rbrace} = \lbrace

\lbrace \rbrace,

\lbrace 10 \rbrace,

\lbrace 20 \rbrace,

\lbrace 10,20 \rbrace

\rbrace \\).

How big is \\(2^S\\)? Well, if \\(S\\) has \\(n\\) elements, the power set \\(2^S\\) has \\(2^n\\) elements:

\\[|2^S| = 2^{|S|}\\]

Hence the notation.