# I don't understand

## Main Question or Discussion Point

I don't understand....

These are things that I have tried looking up but still don't get it. What is..
1. Codomain
2. Morphism(homo, mono, iso, endo, auto, etc.)
3. Transfinite numbers

CRGreathouse
Homework Helper
A codomain is what a function maps into. Take the cardinality function: given a finite set it returns the number of elements. It might be described as mapping $$\mathcal{S}\rightarrow\mathbb{Z}$$ where $$\mathcal{S}$$ is the class of all sets. The range is a subset of the domain; it's the values which the function can actually take. In this case the range is the nonnegative integers.

For another example, $$x\mapsto x^2$$ on $$\mathbb{Z}\rightarrow\mathbb{Z}$$ has domain Z, codomain Z, and range {0, 1, 4, 9, 16, ...}. The range could be equal to the codomain, but it's seldom expressed that way (because mathematicians prefer to use a 'major' or well-known set as the codomain, usually one of the blackboard bold ones. ).

matt grime