Special Types of Sets

A few particular kinds of sets come up so often that they are given their own names.

Empty set (null set): a set with no elements at all, written $\emptyset$ or $\{\ \}$. For example, $\{x : x \in \mathbb{N}, x + 3 = 1\} = \emptyset$, since no natural number added to 3 gives 1. Note that $\{0\}$ is not the empty set — it is a set containing one element, namely $0$; the empty set contains no elements whatsoever.

Singleton set: a set containing exactly one element, such as $\{5\}$ or $\{x : x \in \mathbb{N}, x^2 = 49, x > 0\} = \{7\}$.

Universal set: the set that contains all the objects under consideration in a particular discussion or problem, usually denoted $U$. Every other set discussed in that context is understood to be built from elements of $U$. For instance, if we are discussing the outcomes of rolling an ordinary die, the universal set is $U = \{1, 2, 3, 4, 5, 6\}$, and any set of "favourable outcomes" is automatically a collection drawn from within $U$.

Finite set: a set whose elements can be counted completely, ending at some definite number — that is, the counting process actually terminates. $\{2, 4, 6, 8, 10\}$ is finite (it has exactly 5 elements); so is the empty set (it has 0 elements, so it too is finite). A set like $\mathbb{N}$, where the counting never ends, is called an infinite set — this chapter's own worked problems stay within finite sets, but recognising the finite/infinite distinction matters for the next section on cardinality, which is only defined for finite sets in this syllabus's scope. …