# Homework #3

Due in lecture, Thursday March 21.

Please do the homework in a workbook.

# 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 $$A\cap B\subseteq A-B$$.