The exam will cover chapters 11 - 12 and might have a few concept questions about proof techniques like induction or proof by counterexample, but it you won’t have to write any proofs. Here are problems from the book similar what will be on the exam. You can find the solutions here: chapter 11 solutions and chapter 12 solutions.
What would be a good first sentence of a proof by contrapositive for a claim of the form: “If \(a, b \in A\) are not equal, then \(f(x) \ne f(y)\).”?
Which statement in the following proof is the inductive hypothesis?
Claim. \(\displaystyle\sum_{i = 1}^n k = \frac{n(n+1)}{2}\) for all \(n \in \mathbb{N}\).
Proof. The base case is true since \[\sum_{i = 1}^1 k = 1 = \frac{(1)(2)}{2}.\] Now suppose that the claim is true for any one \(n \in \mathbb{N}\). Then \[\begin{align*} \sum_{i = 1}^{n+1} k &= \left(\sum_{i=1}^{n} k\right) + (n+1) \\ &=\left(\frac{n(n+1)}{2}\right) + (n+1) \\ &=\frac{n(n+1)}{2} + \frac{2(n+1)}{2} \\ &=\frac{(n+1)(n+2)}{2}, \end{align*}\] which completes the proof by mathematical induction. \(\square\)