For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. Notation: Let F be the set of all functions from Z to Z. For all f, g, h = F, if fog=foh, then g = h. For all f, g, h EF, if fog=fo h and f is one-to-one, then g = h.

Let A, B, and C be nonempty sets and let f: A -> B, g: B -> C and h: B -> C be functions. Find a counterexample to disprove the following: If gof = hof, then g = h.

. For nonempty sets A, B and C, let f : A → B and g: B → C be functions. (a) Prove: If g of is one-to-one, then f is one-to-one. using each of the following proof techniques: direct proof, proof by contrapositive, proof by contradiction. (b) Disprove: If go f is one-to-one, then g is one-to-one.

In this problem, assume that A and B are subsets of N, and f is a function N→ N. N-ACN. Write in terms of quantifiers over natural numbers (like the statement in the next part): Write in terms of set operations (like the statement in the previous part): Vn EN,n & A (ne B vn=1). Write in terms of quantifiers: f(A) SA.