Read: End of Section 2.3 and beginning of 2.4.
Turn in: to 2.56, 2.57, 2.59, 2.61, 2.63
- Before starting, make sure to carefully read the text between 2.52 and 2.55. This may be your first formal exposure to proof by contradiction. Definitely come with any questions that come up.
- On 2.56, remember that the book’s definition of $\mathbb{N}$ is all positive integers. Thus, if $x,a\in \mathbb{N}$, then $xa \ge x$ since $a\ge 1$.
- On 2.57, please follow the book’s recommendation to break your proof into two parts: one part to prove $\text{($n$ even)}\implies \text{(4 divides $n^2$)}$ and one part to prove $\text{(4 divides $n^2$)}\implies \text{($n$ even)}$. This first of the two implications should be fairly straightforward with a direct proof; for the second implication, consider trying a proof by contraposition or by contradiction.
Extra practice: 2.55, 2.58
