Mathematics for Computer Science

Chapter 1 What is a Proof?22

(b)Bogus proof: 1 ¢D$0:01D.$0:1/^2 D.10¢/^2 D 100 ¢D$1: 

(c) Bogus Claim: Ifaandbare two equal real numbers, thenaD 0.

Bogus proof.

aDb

a^2 Dab

a^2 b^2 Dabb^2

.ab/.aCb/D.ab/b

aCbDb

aD0:



Problem 1.4.
It’s a fact that the Arithmetic Mean is at least as large as the Geometric Mean,
namely,
aCb
2



p

ab

for all nonnegative real numbersaandb. But there’s something objectionable
about the following proof of this fact. What’s the objection, and how would you fix
it?

Bogus proof.

aCb

2

‹



p

ab; so

aCb

‹

 2

p

ab; so

a^2 C2abCb^2

‹

4ab; so

a^2 2abCb^2

‹

0; so

.ab/^2  0 which we know is true.

The last statement is true becauseabis a real number, and the square of a real
number is never negative. This proves the claim.