Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

3.2. Proofs 219
Example 3.2.7
Prove that √ 2 is irrational.
Solution.
Proof. Suppose not. Then √ 2 is equal to a fraction a b . Without
loss of generality, assume a b
is in lowest terms (otherwise reduce the
fraction). So,
2 a2
b 2
2b 2 a 2 .
Thus a 2 is even, and as such a is even. So a 2k for some integer
k, and a 2 4k 2 . We then have,
2b 2 4k 2
b 2 2k 2 .
Thus b 2 is even, and as such b is even. Since a is also even, we see
that a b is not in lowest terms, a contradiction. Thus √ 2 is irrational.
Example 3.2.8
Prove: There are no integers x and y such that x 2 4y + 2.
Solution.
Proof. We proceed by contradiction. So suppose there are integers
x and y such that x 2 4y + 2 2(2y + 1). So x 2 is even. We have
seen that this implies that x is even. So x 2k for some integer
k. Then x 2 4k 2 . This in turn gives 2k 2 (2y + 1). But 2k 2 is
even, and 2y + 1 is odd, so these cannot be equal. Thus we have a
contradiction, so there must not be any integers x and y such that
x 2 4y + 2.
Example 3.2.9
The Pigeonhole Principle: If more than n pigeons fly into n pigeon
holes, then at least one pigeon hole will contain at least two pigeons.
Prove this!