Download Chapter 1: Logic 10-3-14 §1.4-1.5: Proof by contradiction

Chapter 1: Logic
§1.4-1.5: Proof by contradiction
10-3-14
Remember these definitions:
Definition: Let n and d be integers. We say that d divides n or that n is divisible by d if there
exists an integer k such that n = d · k. If d divides n, we write d|n.
Definition: Let p be an integer. Here are two equivalent ways to say that p is prime:
1. The only divisors of p are 1 and p.
2. If a and b are integers and p|(a · b), then p|a or p|b.
Definition: Let x be a real number. We say that x is rational if there exist integers p and q with
q 6= 0 such that x = pq .
Write proofs of the following statements. In each case, what is the contradiction that you have
reached?
1. We know that 32 + 42 = 52 . Is it possible to find three consecutive integers so that the cube
of the largest number is the sum of the cubes of the other two?
2. If x and y are positive real numbers, then
x
y
+
y
x
≥ 2.
3. Prove that if x is rational and y is irrational, then x + y is irrational.
√
4. In this problem, you will prove that 2 is irrational!!!
√
√
(a) Suppose that 2 is a rational number. This means that 2 = ab for some integers a and
b. We can assume that a and b have no common factors so the fraction ab is reduced to
its lowest terms.
(b) Show that 2b2 = a2 .
(c) Prove that a is even.
(d) Prove that b is even.
√
(e) Conclude that 2 is an irrational number.
5. Prove that if a and b are integers, then a · b + 1 is not divisible by a or b.
6. It is impossible to find prime numbers a, b, and c such that a3 + b3 = c3 . (Hint: Can all three
of a, b, and c be odd? What is prime that is odd (peculiar) because it is the only prime that
isn’t odd?)
7. Here is a question to ponder over the weekend. We have shown that the sum of two rational
√
numbers is rational and the product of two rational numbers is rational. Since 2 is irrational,
it is possible to take two rational numbers, x and y, so that xy is irrational (specifically, x = 2
and y = 21 ). Is it possible to take two irrational numbers, x and y, so that xy is rational?