Mathematics for Computer Science

Part I Proofs

Bogus proof.

aDb

a^2 Dab

a^2 b^2 Dabb^2

.ab/.aCb/D.ab/b

aCbDb

aD0:



Problem 0.2.
It’s a fact that the Arithmetic Mean is at least as large 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. 

Problem 0.3.
Albert announces to his class that he plans to surprise them with a quiz sometime
next week.