So I want to clarify if what I'm thinking is correct.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a mapping f : A → B and we have a in A and b in B.

If f is an injective map, then f(a) = f(b) implies that a = b or conversely a≠b implies f(a)≠f(b).

If f is a surjective map, then for b in B, there exists an a in A such that f(a) = b.

If A = B then f is a homomorphism from A to B if it is operation preserving. That is f(ab) = f(a)f(b) for all a and b in A.

If f is both injective, surjective, and operation preserving, then it is a bijective homomorphism, also known as an isomorphism, and thus has an inverse f^{-1}: B → A.

If f is an injective homomorphism, it is called a monomorphism.

If f is a surjective homomorphism, it is called an epimorphism.

If A = B and f is a homomorphism, then it is called and endomorphism.

Also a bijective endomorphism is an automorphism.

I'm hoping that those are correct ^

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# Clarifying some mapping stuff

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