Power Set and Universal Set

Power set. Given a set $A$, the power set of $A$, written $P(A)$, is the set

of all possible subsets of $A$ — including $\varnothing$ and $A$ itself. Note

carefully that the elements of $P(A)$ are themselves sets, not the original elements

of $A$.

Worked illustration. Let $A = \{1, 2, 3\}$. Its subsets are:

So $P(A) = \big\{\varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\big\}$,

and $n(P(A)) = 8$.

Counting rule. If a finite set $A$ has $n$ elements, then $A$ has exactly $2^n$

subsets, so

This follows because each of the $n$ elements independently has two choices — either

it is included in a given subset or it is not — giving $2 \times 2 \times \cdots \times 2$

($n$ times) $= 2^n$ possible subsets in total. For $A = \{1,2,3\}$ above, $n(A) = 3$

and indeed $2^3 = 8$ matches the count found by listing.

Universal set. When working with several sets at once, it is convenient to fix one

"master" set that contains every element under discussion as a subset; this is called

the universal set, usually denoted $U$. The choice of universal set depends on the

context: if we are discussing sets of natural numbers, we might take $U = N$; if we

are discussing sets of students in a school, $U$ might be the set of all students in

that school. Every set being considered in a given problem is then automatically a

subset of $U$. The universal set plays an essential role later in defining the

complement of a set (Section 1.8) — the complement of $A$ is precisely "everything

in $U$ that is not in $A$," so without first fixing $U$, "complement" would not even

be a well-defined idea. …