A set is a collection of things, which are called the elements of the set.
A one to one correspondence from a set A to a set B is a rule that associates to each element in A exactly one element in B, in such a way that each element in B gets used exactly once and for exactly one element in A.
a function from a set A to a set B is a rule that assigns to each element in A an element of B. If f is the name of the function and a is an element of A then we write f(a) to mean the element of B that is assigned to a. A function f is often written as f: A –>B.
A morphism is a function from A to B that captures at least part of the essential nature of the set A in its image B.
A structure preserving map.
f(x o y) = f(x) o f(y)
A morphism from a source object to a standard target object (morphism of groups).
For instance, systems of equations might be represented as permutation representations or linear representations.
A function from the set X to the set Y. For every y in Y there is exactly one x in X (one to one correspondence). See this page.
A canonical form usually refers to a standard way of simplifying an expression without altering its form - origin obscure.