Types of Sets

Once sets can be described, it is useful to classify them and to describe how one set can sit inside another.

Empty (null) set. A set with no elements at all is the empty set or null set, written $\varnothing$ or $\{\ \}$. For example, $\{x : x \in \mathbb{N},\ 3 < x < 4\}$ is empty, because no natural number lies strictly between $3$ and $4$. Note that $\{0\}$ and $\{\varnothing\}$ are not empty — each contains one element.

Singleton set. A set with exactly one element, e.g. $\{5\}$ or $\{x : x + 2 = 7,\ x \in \mathbb{N}\} = \{5\}$.

Finite and infinite sets. A finite set has a countable, terminating number of elements — the counting comes to an end. An infinite set does not; the counting never ends, e.g. $\mathbb{N}$ or the set of points on a line.

Cardinality. For a finite set $A$, the number of elements in $A$ is called its cardinal number, written $n(A)$. For example, if $A = \{a, e, i, o, u\}$ then $n(A) = 5$.

Equal and equivalent sets. Two sets $A$ and $B$ are equal, written $A = B$, if they have exactly the same elements (so each is a subset of the other). Two finite sets are equivalent if they merely have the same number of elements, i.e. $n(A) = n(B)$. Equal sets are always equivalent, but equivalent sets need not be equal: $\{1, 2, 3\}$ and $\{a, b, c\}$ are equivalent (both have $3$ elements) but not equal.

Subsets. $A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$. If in addition $B$ has at least one element not in $A$ (so $A \ne B$), then $A$ is a proper subset, written $A \subset B$. Two basic facts hold for every set $A$: the empty set is a subset of every set ($\varnothing \subseteq A$), and every set is a subset of itself ($A \subseteq A$).

Universal set. In any particular discussion, the universal set $U$ is the set containing all the objects under consideration; every other set in that discussion is a subset of $U$. For a demographic study, $U$ might be all citizens of a state; for a dice problem, $U = \{1, 2, 3, 4, 5, 6\}$.

Power set. The power set of $A$, written $P(A)$, is the set of all subsets of $A$ (including $\varnothing$ and $A$ itself). If $A$ has $n$ elements, then $A$ has exactly $2^n$ subsets, so: