Functions via set theory
A typical “function” is given by a formula of the form \[ f(x) = \sin(x) \] and we visualize it with its graph:
Functions as (special) relations
The key insight in abstracting the idea of “function” is to understand what the graph of a function really is.
If \(f:A\to B\) is a function, then the graph of \(f\) is the set of points \(G(f)=\{(a,b)\in A\times B: f(a)=b\}\).
Two observations:
- \(G\) is a relation from the set \(A\) to the set \(B\) since \(G\subset A\times B\).
- Everything we need to know about \(f\) is stored in \(G\).
\(A\) is called the domain of \(f\). \(B\) is called the codomain of \(f\).
Functions as (special) relations continued
The key property that makes a general relation a function is the fact that
for all \(a\in A\), there exists a unique \(b\in B\) so that the pair \((a,b)\in G(f)\). (note the quantifiers here).
Notice that for a general relation, there is no such condition – any subset \(R\) of \(A\times B\) is a relation.
A general relation vs a function
Drawn in this way, a relation \(R\subset A\times B\) is a function if it passes the vertical line test - every vertical line hits exactly one point in \(B\).
relations vs functions continued
We can also explore the special properties of functions among relations using the other way of representing functions.
The range of a function
Definition: The range of a function \(F\) is the set of \(b\in B\) such that there exists \(a\in A\) with \((a,b)\in F\).
In “old fashioned” terms, the range of \(F\) is the set of \(b\) for which there exists \(a\) with \(F(a)=b\).
Example of the range of a function
(Example 12.3 from the book). We define \(\phi:\mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}\) by the formula \(\phi(m,n)=6m-9n\). As a set, this is the function \(\{(m,n),6m-9n\}\) as a subset of \(\mathbb{Z}^2\times\mathbb{Z}\).
What is its range?