1. What is a function

What is a function?

Function

Let $A$ and $B$ be any sets. A function from $A$ to $B$ is a rule $f$ that associates each element of $A$ with exactly one element of $B$.

If $f$ associates $x\in A$ with $y\in B$, then we write $\boxed{f(x)=y}$ and say ”$f$ of $x$ is $y$” or ”$f$ maps $x$ to $y$” or the “value of $f$ at $x$ is $y$”.

If $y$ is a function from $A$ to $B$, then we write: $\boxed{f:A\to B}$.

Function Rules

We call $A$ the domain of $f$ and $B$ the codomain of $f$.

• every element of the domain has to be mapped somewhere in the codomain (but this does not have to apply in reverse)
• one element cannot be mapped to 2 different places (but it can happen that $2$ different elements are mapped to the same place)

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Example 1

Let $H$ be the set of all humans, alive or dead. Making $H\to H$ associations, are the following functions or not:

• $\boxed{f(x)\text{ is a parent of }x}$ Not a function: people have two parents
• $f(x)\text{ is the mother of }x$ Function: each person has exactly one mother
• $f(x)\text{ is the oldest child of }x$ Not a function: some people have no children
• $f(x)\text{ is the set of all children of }x$ This is a function but it is not a $H\to H$ function because each person is associated with a set of people. This $f$ is a $H\to P(H)$ function.