CS 230 : Discrete Computational Structures
Homework Assignment #3
Due Date: Monday, February 22
Suggested Reading: Rosen Sections 1.7 - 1.8; Lehman et al. Chapter 1
For the problems below, explain your answers and show your reasoning.
1. [5 Pts] Prove, using a direct proof that p is odd if and only if p
3
is odd.
2. [6 Pts] Let x and y be non-zero rational numbers and let z be an irrational number.
Prove that x + yz is irrational. Can you use a direct proof? Why or why not?
3. [6 Pts] Let m and n be positive integers. Prove, by contrapositive, that if mn > 35,
then m ≥ 6 or n ≥ 8.
4. [6 Pts] Suppose your college organization has 32 students. Prove that it has at least
5 freshmen or at least 8 sophomores or at least 10 juniors or at least 7 seniors.
5. [6 Pts] Prove by cases that if p ≥ 3 or p ≤ −7 then (p + 2)2 ≥ 25.
6. [6 Pts] Prove that the square root of 5 is irrational.
7. [5 Pts] Prove that there exist rational numbers x and y where x
y
is irrational. Is your
proof constructive or non-constructive? Explain.
For more practice, work on the problems from Rosen Sections 1.7 - 1.8 and LLM Chapter 1.
