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 \(A\cap B\subseteq A-B\).