functions with Domains Larger than codomain

Question

I wanted to know if "functions" with codomains smaller than their domains are not funtions:

for example: R-> Z f(x)=|x|

is not a function since where are many real numbers that are not transformed to an interger value (I think).

Answer 1

The problem is not that the domain is larger than the codomain, it is that some values of $x \in \Bbb R$ do not have an image. $g(x): \Bbb {R \to Z}\ g(x)=1$ is a perfectly good function with the same domain and codomain as your example.

Answer 2

In set theory a function $f$ is by definition a set of ordered pairs with a special property.

Its domain is the set $X:=\{x\mid\langle x,y\rangle\in f\}$ and the special property states that for every $x\in X$ there is exactly one $y$ such that $\langle x,y\rangle\in f$.

Its range (or image) is the set $\{y\mid\langle x,y\rangle\in f\}$.

For its codomain you can choose every set $Y$ that contains its range as a subset.

Special example: $f=\{\langle x,0\rangle\mid x\in\mathbb R\}$ where $\mathbb R$ serves as domain and where $\{0\}$ is the range of $f$.

As codomain we can choose any set that contains $0$.

For example $\{0\}$ which is apparantly smaller than domain $\mathbb R$.