Homework 1
Please note that after this homework, you will need a copy of the textbook to get
the questions. I only typed these up since some students are waiting for their textbook
that they ordered.
Section 1.1
2. Which of these are propositions? What ar
CS 1050 B: Construction Proofs
January 25, 2007
Solutions to Homework 1
Lecturer: Sasha Boldyreva Problem 1.1, 4 points. (There was a typo here.) Prove that the proposition "if it is not possible to solve the problem, then it is possible to solve i
Theorem 2.5. There are infinitely many prime numbers. Proof. Let p be the proposition
that there are infinitely many prime numbers. Suppose there are only finitely many
primes (that is, assume p to be true). This would imply that any positive integer > 1
NOTE: This homework is due Wednesday at the start of class because of the exam on Friday!
Please note that these problems come from the sixth edition of the United States version of Discrete Mathematics and Its Applications by Kenneth Rosen. The internati
A Guide to Proofs by Contradiction
Ashwin Bhat
1
What is a proof by contradiction?
A proof by contradiction (also known as reduction ad absurdum ) is a proof technique in which a proposition
is shown to be true by showing that if the proposition were fals
Hai Tang (htang32) Lisa Thornsberry (lthornsberry3) Kevin Jones (kjones41)
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Tingyu Zhu (tzhu7) Rebecca Williams (rwilliams7) Charles Nomides (cnomides3)
Rebecca Riles (rriles3) Jenny Kim (
CS1050
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[M Z] + Hor
Trey Kester
Nick Wood
(If your last name begins wit
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Please note that these problems come from the sixth edition of the United States version of Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of
Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may
not contain the same problems in the same places so be sure that you are
Please note that these problems come from the sixth edition of the United States version of Discrete Mathematics and Its Applications by Kenneth Rosen. The international edition may not contain the same problems in the same places so be sure that you are
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Theorem 2.3. If n 2 is even, then n is even. Which type of proof by contradiction should
we use here? The important thing to notice is that this is a proposition of the form If p,
then q, so we are trying to show that p q. Typically when the proposition i
What is a proof by contradiction?
A proof by contradiction (also known as reduction ad absurdum) is a
proof technique in which a proposition is shown to be true by showing that
if the proposition were false, a logical contradiction would arise. This is th
We have shown that s is the difference of two rational numbers, which, by the
definition of the set of rational numbers, is itself a rational number (if youre curious
about this, read about group theory). This contradicts our assumption of the proposition
Example Proofs Using these definitions, lets prove a fact regarding irrational and
rational numbers using a proof by contradiction. Theorem 2.1. The sum of a rational
number, r, and an irrational number, s, is an irrational number. Proof. Well prove this
The contradiction here is a bit harder to see than in the previous question. Looking
closer, we see that we have just shown that c 2 > a2 + b 2 . Let c 2 6= a 2 + b 2 be
proposition r. Upon reexamining the initial proposition that we are trying to prove,
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CS 1050A: Constructing Proofs
Solutions to Quiz 1
Problem 1 (20 points)
For this problem, you can assume that 2 is irrational (which we proved in class).
2 x + (2- 2) y . 2
Let x and y be two rational numbers such that x < y. Let z = 1. Prove that x < z.