What Are Codomains, Morphisms, and Transfinite Numbers in Mathematics?

[Skhandelwal]

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]

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]

1. Done.
2. A map. Homo means structure preserving, iso means invertible, endo means the domain and codomain are the same, i.e. a map f:X-->X, and auto is an invertible endo. All of these can be found simply by using google, by the way. If you have any more 'what is the definition' questions your first step should always be to use google to look them up (mathworld is a useful online resource). Of course, if you mean 'I have the definition but don't understand it' then it would be better if you wrote out what you think the definition is so someone can explain it to you.
3. The cardinals come with an arithmetic. The transfinite ones are those that are not cardinals of finite sets. E.g. Alpeh_0, c the cardinality of the continuum.