10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 343- This exercise relates to the proof of Theorem 7.
(a) Prove Theorem 7(b); that is, 1. n = n 1 = n for all n E N.
(6) Prove Theorem 7(c); that is, multiplication on N is commutative.
(c) Prove Theorem 7(e); that is, multiplicative cancellation on N. - (a) Prove Theorem 8.
(b) Prove Theorem 9, parts (a), (c), and (d). - Assume a, b, c, d E N. Prove:
*(a) If a + c < b + c, then a < b
(b) If ac < bc, then a < b. [Note the relationship between the results in (a) and
(b) and parts (g) and (h) of Theorem 8. Hint for the proof: Use Theorem 10, that
is, trichotomy.]
(c) If a < b and c < d, then a + c < b + d
(d) If a < b and c < d, then ac < bd
- Define a relation 5 on N by the rule a < b if and only if either a < b or a = b.
Prove that, given a, b E N:
(a) a I a (i.e., I is a reflexive relation on N)
(6) a < b if and only if a I b and a # b
(c) If a 5 b and b 5 a, then a = b (i.e., I is an antisymmetric relation on N)
(d) If a ( b and b 5 c, then a I c (i.e., I is a transitive relation on N)
(e) Either a I b or b I a
(f) If b < a, then b + 1 I a
(g) 1 is the least element of N
[Note: If you have read Article 7.4, you will recognize that (N, I) is a partially
ordered set, or poset, by parts (a), (c), and (d), and indeed is a totally ordered set,
due to (e).] - *(a) Write a direct proof of the principle of mathematical induction (IP) from
the well-ordering principle.
(b) Use IP2 to prove that any set X of n real numbers has a smallest element.
10.2 Development of the Integers
and Rational NumbersTHE INTEGERS
As noted in Article 10.1, the system N of positive integers satisfies all the
field axioms except the additive identity, additive inverse, and multiplicative
inverse properties. We will soon see that we can construct from N, by
means of equivalence classes, the number system Z of all integers, including
negative integers and zero. If the mathematical object about to be con-
structed corresponds to the integers, as we know them intuitively, we should
expect the additive deficiencies of N to be remedied in Z.
As suggested earlier, it is at this point that equivalence classes begin to
assume a crucial role. We begin our development by considering the set