Advanced Power Set Calculator

The power set of a set S, denoted as \(P(S)\), is the set of all possible subsets of S, including the empty set (\(\emptyset\)) and the set S itself.

For example, if \(S = \{a, b\}\), its subsets are:

• \(\emptyset\) (the empty set)
• \(\{a\}\)
• \(\{b\}\)
• \(\{a, b\}\)

So, the power set is \(P(S) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}\).

Size of a Power Set:

If a set S has \(n\) elements (its cardinality is \(|S| = n\)), then the size of its power set is \(|P(S)| = 2^n\). This is because for each element in S, there are two choices: either include it in a subset or not.