From the Text

Complete the following questions from the text:

• Section 1.6: 12.
• Section 1.7: 12, 20.
• Section 2.1: 8, 18, 20, 30.
  • For 20, give a proof.
• Section 2.2: 4

Questions

1. In 1.6-12, the question specified “nonzero rational number” what in your proof would have failed if the rational number was zero? (It must fail: $0\cdot\sqrt{2}$ is rational.)
2. Prove that $x$ is rational if and only if $x+1$ is rational.
3. According to Wikipedia, there are 39000 students at ZJU. Show that on one day, at least 107 of them must have a birthday.
4. Prove that for an integer $n$, either 3 divides $n$ or 3 divides $n^2−1$. [Hint: by cases $n=3k, 3k+1, 3k+2$ for some integer $k$.]
5. Prove or disprove that for sets $A$ and $B$, it is always the case that $A-B\in P(A)$.
6. Prove or disprove that for sets $A$ and $B$, it is always the case that $A\cap B\subseteq A-B$.