Math 310 Collected Homework #2
Due Friday, 10/10/08
Proofs (direct, by contradiction, by induction)
Write clear, neat, and complete proofs. Label the start of each proof by โProof:โ, and end with
QED or โก. Do not include your sketches or scratch work. Please leave a little space around your
proof for comments.
1. Write a direct proof to show that for all integers ๐, ๐๐ + ๐ is even .
2. Let m and n be two arbitrary integers. Show that the product mn is odd if and only if
both m and n are odd.
3.
a) Give a counterexample to show that ๐ โค ๐ => ๐๐โค ๐๐ is not always true.
b) Prove that for all positive real numbers ๐,๐: ๐ โค ๐ => ๐ โค ๐
4. Prove that if x is a rational number and y is an irrational number, then their sum x+y is
an irrational number.
(Recall & use the definitions:
๏ท x is a rational number if ๐ฅ=๐
๐ for some integer numbers a and b, where ๐ โ 0.
๏ท A real number x is irrational if it is not a rational number)
5-7. From the Problem Set I (textbook, pages 53-57) do problems #11, #12, and #16.
8. From the Problem Set I (textbook, pages 53-57), problem #10