Power set

Definition: Power set
The power set of a set (written )is the set of all subsets of . Formally, .

Because the elements of the power set are themselves sets, it is easy to confuse elements and subsets. As with any set comprehension, the elements of are the subsets of . Here are some examples:

Let . Then

• , by inspection. For example, because .
• , because 1 is a number but contains only sets. Put differently, .
• Therefore
• But note that .
• for any set , we have and , because ∅⊆X and X⊆X.