There was a typo in the assignment that I handed out. The exercise mentioned in Problem 4 should be Exercise 2.9.7. The corrected version you can read here.
Read Section 2.8. Understand the definitions and all the statements. This section consists of three parts: the statements about finite sets, the statements about ${\mathbb N}$ and the statements about countable sets. You should do at least one proof in each part.
The most important exercise about finite sets is: 2.8.6. (5 points for a successful presentation). Mathematical induction is a natural way to proceed here. It is essential to be specific what is the statement $P(n)$ that is being proved.
Notice that I proved Exercise 2.8.15: Every nonempty subset of ${\mathbb N}$ has a minimum.
Abstract proofs about countable sets are somewhat technical. Using what is proved in the notes, you should be able to do 2.8.21. As always, it is essential to distinguish between green and red parts of the proof. Then one can develop a strategy to construct red parts using green.
Exercise 2.2.11. Let $\alpha \in {\mathbb R}.$ Prove that $\alpha < x$ for all $x > 0$ implies $\alpha \leq 0.$ The proof is simple. Just state the contrapositive. If $\alpha > 0,$ then there exists $x > 0$ such that $\alpha \geq x.$ The contrapositive is clearly true since we can take $x = \alpha.$
Exercise 2.2.11a. Let $\alpha \in {\mathbb R}.$ Prove that $\alpha \leq x$ for all $x > 0$ implies $\alpha \leq 0.$ The proof is again simple. Consider the contrapositive. If $\alpha > 0,$ then there exists $x > 0$ such that $\alpha > x.$ Again, the contrapositive is true since we can take $x = \alpha/2.$