A well defined function.

Question

I don't understand how the following statement implies a function is well defined:

For $$ f: X \rightarrow Y$$

'A function $f$ is well defined rule that assigns a unique element $f(x) \in Y$ to each $x \in X$.

Answer 1

It is just a definition.

A function $f:X\to Y$ is not well defined (so it is actually wrong to call it a function) if one of the two following things happen:

• (1) there exists $x\in X$ such that $f(x)$ does not exist,

• (2) there exists $x\in X$ such that $f(x)$ correspond to more than one element of $Y$.

For instance,

$$f:x\mapsto 1/x$$

is not well defined on $\mathbb R$ because of (1).

And

$$f:\mathbb Z/2\mathbb Z\to \mathbb Z$$

defined by $f(\bar a)=a$ (where $\bar a$ is the projection of $a\in\mathbb Z$ in $\mathbb Z/2\mathbb Z$) is not well defined because $\bar 0=\bar 2$ but $f(\bar 0)=0\ne 2=f(\bar 2)$ (because of (2)).

Answer 2

A function is well-defined $f: X\to Y$ when every element in the domain $x\in X$ is mapped to one and only one element in the range $f(x)\in Y$.

Examples of a non-well defined "function" would be the following

1. $f(\frac{m}{n})=\frac{1}{n}$ where $f: \mathbb{Q}\to \mathbb{Q}$. Notice that this isn't well defined because $f(\frac{1}{2})\neq f(\frac{2}{4})$.( Note that this can be fixed by requiring that $\frac{m}{n}$ be a reduced fraction.)

2. $f(x)=\frac{1}{x}$ where $f:\mathbb{R}\to \mathbb{R}$ which isn't defined at $0$.(note that this can be fixed if the domain is restricted to $\mathbb{R}/\{0\}$)

In both of these examples our "Non-well defined" functions aren't actually functions at all. For a rule to be a function it must be well defined.