Problem 3
Let f(x)= 3x/2 . For this problem, we will use the notation cZ to denote the set of multiples of c.
For example, 2Z denotes the even integers, and 3Z denotes multiples of 3.
(a) Prove that f: 2Z → Z is one-to-one via a direct proof.
Recall that we did an example of such a proof in the second lecture on functions; use that
proof to help structure your proof.
Grading Notes. While a detailed rubric cannot be provided in advance as it gives away
solution details, the following is a general idea of how points are distributed for this problem.
We give partial credit where we can.
(3) Correctness. If your proof is not correct, this is where you'll get docked. You'll need
(1) the definition of one-to-one in the "forwards" direction,
(2) other facts to prove f is one-to-one.
(2) Communication. We need to see a mix of notation and intuition, preferably in the
"column" format with the notation on the left, and the reasons on the right. If you
skip too many steps at once, or we cannot follow your proof, or if your solution is overly
wordy or confusing, this is where you'll get docked.
(b) Prove that f: 2Z → Z is one-to-one via a proof by contrapositive.
Grading Notes. While a detailed rubric cannot be provided in advance as it gives away
solution details, the following is a general idea of how points are distributed for this problem.
We give partial credit where we can.
(4) Correctness. If your proof is not correct, this is where you'll get docked. You'll need
(1) a statement that the proof is proceeding by contrapositive,
(1) the definition of one-to-one in its contrapositive form,
(2) other facts to prove f is one-to-one.
(2) Communication. We need to see a mix of notation and intuition, preferably in the
"column" format with the notation on the left, and the reasons on the right. If you
skip too many steps at once, or we cannot follow your proof, or if your solution is overly
wordy or confusing, this is where you'll get docked.
(c) Decide whether f: 4Z→ 3Z is one-to-one and prove your claim.
