In this section, we’ll seek to answer the questions:
What is a function?
How can functions be represented?
What special properties can functions possess?
In this section, we explore functions. As with everything in this class, we want to take our intuition from continuous mathematics (college algebra, calculus)and extend it to more general situations. Moreover, our work from the previous section allows us to use our knowledge of relations to compactly state the definition of a function.
Definition5.2.1.
Let \(X\) and \(Y\) be nonempty sets. A function from \(X\) to \(Y\) is a relation \(f\) such that:
For all \(x\in X\) there exists \(y\in Y\) such that \((x,y)\in f\text{,}\) and
If \((x,y), (x,z)\in f\) then \(y = z\text{.}\)
We call \(X\) the domain of \(f\) and \(Y\) the codomain of \(f\text{,}\) and often write \(f : X\to Y\) to mean “\(f\) is a function with domain \(X\) and codomain \(Y\)”. Given \(x\in X\) such that \((x,y)\in f\text{,}\) we generally write \(y = f(x)\) to mean that \(y\) is the image of \(x\) under \(f\).
LaTeX Code5.2.2.
To typeset \(f: X\to Y\) in \(\LaTeX\text{,}\) use f: X \to Y in math mode.
Activity5.2.3.
Describe the condition imposed by the second bullet point in Definition 5.2.1. What was this called in, say, a college algebra/precalculus class?
Definition5.2.4.
Let \(f : X\to Y\) be a function.
Let \(A\subseteq X\text{.}\) The image of \(A\) is the set \(f(A) = \setof{f(x)}{x\in A}\text{.}\)
The image of \(X\) is known as the range of \(f\), and is denoted \(\ran(f) = f(X)\text{.}\)
Given a set \(B\subseteq Y\text{,}\) the preimage of \(B\) under \(f\), denoted \(f^\leftarrow (B)\text{,}\) is the set \(f^\leftarrow(B) = \setof{x\in X}{f(x)\in B}\text{.}\)
Activity5.2.5.
Describe in words the range, image, and preimage of a function. Use as few symbols as possible.
Problem5.2.6.
Let \(X = \set{a,b,c,d,e}\) and \(Y = \set{1,2,3,4,5}\text{.}\) Define a relation \(f\) from \(X\) to \(Y\) by
\begin{equation*}
f = \set{(a,3), (b,3), (c,1), (d,2), (e,5)}.
\end{equation*}
Does \(f\) define a function from \(X\) to \(Y\text{?}\) Explain.
What is the range of \(f\text{?}\) Does the range equal the codomain?
Let \(S = \set{a,b,d}\) and calculate \(f(S)\text{.}\)
Let \(T = \set{1,4,5}\) and calculate \(f^\leftarrow(T)\text{.}\)
Let \(g\) be the relation from \(Y\) to \(X\) given by
\begin{equation*}
g = \setof{(y,x)}{(x,y)\in f)}.
\end{equation*}
Is \(g\) a function? Explain.
When proving theorems about images, preimages, etc., we refer to the definitions as usual.
Theorem.
Let \(f : X\to Y\) be a function with \(S\subseteq T\subseteq X\text{.}\) Then \(f(S)\subseteq f(T)\text{.}\)
Proof. Let \(y\in f(S)\text{.}\) This means there exists \(x\in S\) such that \(f(x) = y\text{.}\) Since \(x\in S\) and \(S\subseteq T\text{,}\)\(x\in T\text{.}\) By definition, \(y = f(x) \in f(T)\text{.}\) Since \(y\) was arbitrary, we conclude that \(f(S)\subseteq f(T)\text{.}\)
In the problems that follow, your goal is to either prove the assertion using standard tools of set theory and the definitions above, or disprove the assertion and salvage it. That is, perhaps the two sets are not equal, but one is always a subset of the other; prove that.
Problem5.2.7.
Let \(f : X\to Y\) and \(S,T\subseteq X\text{.}\) Then \(f(S\cup T) = f(S)\cup f(T)\text{.}\)
Problem5.2.8.
Let \(f : X\to Y\) and \(S,T\subseteq X\text{.}\) Then \(f(S\cap T) = f(S)\cap f(T)\text{.}\)
Problem5.2.9.
Let \(f : X\to Y\) and \(S,T\subseteq X\text{.}\) Then \(f(S\setminus T) = f(S)\setminus f(T)\text{.}\)
Problem5.2.10.
Let \(f : X\to Y\) and \(U\subseteq V\subseteq Y\text{.}\) Then \(f^\leftarrow (U) \subseteq f^\leftarrow (V)\text{.}\)
Problem5.2.11.
Let \(f : X\to Y\) and \(U, V\subseteq Y\text{.}\) Then \(f^\leftarrow (U\cup V) = f^\leftarrow (U)\cup f^\leftarrow (V)\text{.}\)
