Types of Sets

This section catalogues the standard vocabulary used to classify sets.

EMPTY SET: a set containing no element at all is called an empty set or null set, denoted by the symbol $\phi$ or $\{\}$ (also called the void set). For example, $A=\{x/x\in N, 1<x<2\}=\{\}$, since no natural number lies strictly between 1 and 2; here $n(A)=0$.

SINGLETON SET: a set containing exactly one element. For example, if A is the set of all integers that are neither positive nor negative, then $A=\{0\}$, and $n(A)=1$.

FINITE SET: the empty set, or any set containing a finite number of objects, is called a finite set. For example, the set of letters in the word 'BEAUTIFUL' is $A=\{B,E,A,U,T,I,F,L\}$ with $n(A)=7$ (note: repeated letters like the second U are listed only once) — A is a finite set.

INFINITE SET: a set that is not finite. Examples include the set of natural numbers, the set of rational numbers, or the set of all points on a circle. Note that an empty set is always considered finite (it is the smallest possible finite set, with 0 elements).

SUBSET: a set A is said to be a subset of a set B if every element of A is also an element of B, written $A \subseteq B$. Two useful facts: $\phi$ is a subset of EVERY set, and every set is a subset of itself ($A \subseteq A$).

SUPERSET: if $A\subseteq B$, then B is called a superset of A, written $B \supseteq A$.

PROPER SUBSET: a nonempty set A is a proper subset of B if every element of A is in B AND at least one element of B is not in A — i.e. $A\subseteq B$ and $A \ne B$, written $A \subset B$. For example, if $A=\{1,3,5\}$ and $B=\{1,3,5,7\}$, then every element of A is in B, but $A\ne B$ (since 7∈B but 7∉A), so $A\subset B$. If even a single element of A fails to be in B, then A is NOT a subset of B at all, written $A \not\subset B$.

POWER SET: the set of ALL subsets of a given set A is called the power set of A, denoted $P(A)$ — every element of a power set is itself a set. For example, if $A=\{a,b\}$, its subsets are $\phi, \{a\}, \{b\}, \{a,b\}$ (four subsets in all), so $P(A) = \{\phi, \{a\}, \{b\}, \{a,b\}\}$. In general, if $n(A)=m$, then $n[P(A)]=2^m$.

EQUAL SETS: two sets are equal if they contain exactly the same elements, i.e. $A\subseteq B$ and $B\subseteq A$. For example, if X is the set of letters in 'ABBA' and Y is the set of letters in 'BABA', then $X=\{A,B\}$ and $Y=\{B,A\}$ — the same two letters — so $X=Y$. …