Why Read This Book? - Index of

EXERCISE 3.10.2 √ 2 is irrational. 33,34,35,36,37
3.11 Relations in General 111
A few words are in order about irrational numbers. Let’s work backwards from
Exercise 3.10.2. Exercise 3.10.2 says √ 2 is not rational, while Theorem 3.9.1 says
√ 2 is real. Thus there do, in fact, exist real numbers that are irrational. But
Theorem 3.9.1 is based on assumption A22, which we have said very little about.
As we said, assumption A22 is not an axiom of the real numbers. It can be proved
from the Least Upper Bound axiom, which is a standard axiom of the real numbers.
So what we discover is that the irrational numbers owe their existence to the Least
Upper Bound axiom.
EXERCISE 3.10.3 The proof of Exercise 3.10.2 can be easily generalized to
demonstrate that √ p is irrational for any prime number p. Explain how your
proof of Exercise 3.10.2 can be adapted into a proof of this more general claim. 38
EXERCISE 3.10.4 The sum of a rational and an irrational is irrational. 39
EXERCISE 3.10.5 The product of a nonzero rational and an irrational is
irrational.
EXERCISE 3.10.6 Let a, b, c, and d be rational numbers, and suppose a + b √ 2 =
c + d √ 2. Does it follow that a = c and b = d? 40
3.11 Relations in General
Equivalence relations are just one example of the more general mathematical
idea of a relation. A relation is a set construction that puts all kinds of element
comparisons such as equality, less than, divisibility, and subset inclusion
into one mathematical idea. It is also a way of linking elements of two different
sets together, and it is a context in which functions can be defined. In this section,
we define a relation and look at examples and special kinds of relations. But first,
we define the Cartesian product of two sets A and B by
A × B ={(a, b) : a ∈ A, b ∈ B} (3.64)
33 Naturally, you will want to assume √ 2 is rational and arrive at a contradiction.
34 Write √ 2 = m/n, where you can assume no common factors between m and n.
35 Square both sides and look at the contrapositive of Corollary 2.4.5 .
36 If m 2 is even then, ....What does this mean? Cancel out a 2.
37 So n 2 is even. What contradiction does this cause?
38 Corollary 2.5.12 might help.
39 See Exercise 1.2.18(i).
40 If b �= d, then what must be true of √ 2?