(i) What does it mean to say that a function is injective? What does it mean to say that is surjective? Let be a function. Show that if is injective, then so is , and that if is surjective, then so is .
(ii) Let be two sets. Their product is the set of ordered pairs with . Let (for be the function
When is surjective? When is injective?
(iii) Now let be any set, and let be functions. Show that there exists a unique such that and .
Show that if or is injective, then is injective. Is the converse true? Justify your answer.
Show that if is surjective then both and are surjective. Is the converse true? Justify your answer.