CMPS 211 Homework Assignment 5, Fall 2014
American University of Beirut · Discrete Structures · 2 pages
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American
University
of
Beirut
Department
of
Computer
Science
CMPS
211
–
Discrete
Mathematics
–
Fall
14/15
Please solve the following exercises and submit BEFORE 8:00 am of
Tuesday 14th, October.
Exercise 1
(15 points)
a) Prove that if a + b and b + c are odd integers where a, b and c are integers, then a
+ c is even. What kind of proof did you use?
b) Prove that if a and b are integers and ab is odd then both of a and b are odd. What
kind of proof did you use?
Exercise 2
(15 points)
a) Prove by contradiction that the sum of an irrational number and a rational number
is irrational.
b) Prove or disprove that the product of a nonzero rational number and an irrational
number is irrational.
Exercise 3
(10 points)
Prove that the proposition P(1), where P(n) is the proposition “If n is a positive integer
then 2n ≥ n+1” is true. What kind of proof did you use?
Exercise 4
(10 points)
Prove that if n is a perfect cube, then n+3 is not a perfect cube.
Exercise 5
(10 points)
Prove that if n is a positive integer, then n is odd if and only if 3n + 2 is odd.
Exercise 6
(10 points)
Is this reasoning for finding the solutions of the equation √(5x2 − 4)=2x correct? (1)
√(5x2−4)=2x is given; (2) 5x2 − 4 = 4x2, obtained by squaring both sides of (1); (3) x2 −
4 = 0, obtained by subtracting 4x2 from both sides of (2); (4) (x − 2)(x + 2) = 0, obtained
by factoring the left-hand side of x2−4; (5) x=2 or x=−2, which follows because ab = 0
implies that a = 0 or b=0.
American
University
of
Beirut
Department
of
Computer
Science
CMPS
211
–
Discrete
Mathematics
–
Fall
14/15
Exercise 7
(10 points)
Prove that there is a positive integer that equals the sum of the positive integers not
exceeding it. Is your proof constructive or non-constructive?
Exercise 8
(10 points)
Show that these statements about the real number x are equivalent:
1. x is rational,
2. x/3 is rational,
3. 5x – 2 is rational.
Exercise 9
(10 points)
Show that these statements about the integer n are equivalent:
1. n2 is even,
2. 1-n is odd,
3. n3 is even,
4. n2+ 1 is odd.