Problem 2 (15 pts.) Recall the following definitions from lecture about a function g : A - B: one-to-one: Vn, me A : (n * m) = (g(n) # g(m)) onto: Vb E B : Ha E A : g(a) = b. Let f : N -+ Z be defined by f(n) := EvEK, deg(v), where Kn is the complete graph on n nodes. (a) (1 pt.) Suppose you are trying to prove a statement of the form Vr E S : [P(x) = Q(x)]. What is the first line of this "for all" proof, as we've seen in this course? (b) (1 pt.) Suppose you are trying to prove a statement of the form P(x) = Q(x). What is the contrapositive of this claim? (c) (1 pt.) Suppose you are trying to prove a statement of the form P(x) = Q(x). Fill in the blanks with a correct structure of a proof by contrapositive of this statement. To prove P(x) = Q(x) by contrapositive, we assume and use that to show (d) (1 pts.) Suppose you want to prove that f : N - Z is one-to-one. What is the statement of what you want to show, given the given definition of one-to-one and this function's domain and codomain?(e) (8 pts.) Fill out the following proof with a proof by contrapositive that f : N ) Z is onetoone. You may use that (n m) (n + m - 1) = n2 - n - m2 + m if it helps. Be sure you are using the contrapositive of the denition of one-toone above, as we did in lecture, tutorial, and PS5. Use parts (a)(d) to help structure your proof! Proof. We want to show (your answer 'om (d)) To prove this statement, (you: answer from (a) applied to (d)) We will prove the equivalent contrapositive, that is, (your answer from (b) applied to (d)). To do this, we will assume and use that to show (your answers from (c) applied to (d)). Reason this Statement is Time Mathematical Statement (From the Approved List) (this should be the second blank from (c) applied to (d)) mumm- (f) (3 pts.) Prove that f is not onto using a disproof by counter-example. Be sure to clearly state the counter-example and precisely demonstrate why it disproves onto