In What Sense Is a Function a Set?

A function is often written as a rule, such as $f(x)=x^2$, or as an arrow from one set to another. In set theory, the rule can be represented by a set of ordered pairs: one pair for each input and its output. This set is called the function’s graph.

That representation explains in what sense functions are sets. It also highlights a convention: the graph alone records which outputs occur, but it does not always specify the codomain named when the function is introduced.

The graph of a function

Suppose $f$ is a function with domain $A$ and codomain $B$. Its graph is

$$\operatorname{Graph}(f)=\{(x,f(x)):x\in A\}.$$

Each member is an ordered pair: its first entry is an input and its second entry is the corresponding output. Since every output lies in $B$, the graph is a subset of the Cartesian product:

$$\operatorname{Graph}(f)\subseteq A\times B.$$

Not every subset of $A\times B$ is the graph of a function from $A$ to $B$. A function’s graph must contain exactly one pair whose first entry is $x$, for every $x\in A$. In particular, a relation that assigns two different outputs to the same input is not a function.

A worked example

Let $A=\{1,2,3\}$, let $B=\{1,4,9,16\}$, and define $f:A\to B$ by $f(x)=x^2$. Evaluate the rule at each input:

$$f(1)=1,\qquad f(2)=4,\qquad f(3)=9.$$

The graph is therefore

$$\operatorname{Graph}(f)=\{(1,1),(2,4),(3,9)\}.$$

It is a set whose elements are ordered pairs, and it is a subset of $A\times B$. Notice that $16$ is in the stated codomain but is not an output. A function does not have to use every member of its codomain.

By contrast, $\{(1,1),(1,4),(3,9)\}$ is not the graph of a function with domain $A$: the input $1$ is paired with two outputs, and $2$ has no output.

How are ordered pairs sets?

Set theory can define an ordered pair using sets. One standard encoding is the Kuratowski pair:

$$(a,b)=\bigl\{\{a\},\{a,b\}\bigr\}.$$

This encoding is chosen so that the pair’s order can be recovered: $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. Other encodings are possible; what matters is that ordered pairs have this distinguishing property.

With ordered pairs encoded as sets, a graph is a set of sets. The function is not a separate kind of object that set theory cannot describe: it can be represented entirely using sets. In ordinary mathematics, people usually write $(a,b)$ rather than expand the encoding, because the notation is much easier to read.

Does the graph include the codomain?

There are two common ways to define a function.

1. As a graph: A function is a set of ordered pairs satisfying the function condition. Its domain is the set of first entries, and the outputs are the set of second entries.
2. As a mapping with specified sets: A function from $A$ to $B$ is recorded as $(A,B,G)$, where $G$ is its graph and $G\subseteq A\times B$.

The distinction matters because the same graph can be paired with different codomains. In the example, the graph also defines a function from $A$ to $\{1,4,9\}$, as well as one from $A$ to $B=\{1,4,9,16\}$. The outputs have not changed, but the codomain has. If a problem asks whether two functions are equal, check whether its convention counts the codomain as part of the function.

The graph determines the domain and all input-output assignments. It does not determine a larger codomain that includes values never attained. The set of actual outputs, called the range or image, is determined by the graph.

Common mistakes

• Treating the formula as the whole function. The expression $x^2$ does not by itself specify a domain or codomain. Those sets can affect which mapping is meant.
• Confusing range with codomain. The range consists of outputs actually attained; the codomain is the target set specified for the function. They may differ.
• Thinking any set of pairs is a function. Each domain element must occur as a first entry exactly once. Missing inputs or repeated inputs with different outputs violate the function condition.
• Assuming every function is a set in every context. In standard set theory, a function with a set domain and set-valued outputs has a set graph. A function defined on all sets, such as the assignment sending each set to itself, is instead a proper-class-sized mapping in common set-theoretic foundations, not a set of all its pairs.

For most elementary problems, the practical answer is simple: view a function as the set of all input-output pairs in its graph, and keep the domain and codomain explicit when they matter.