Homework 1
Assignment: This is due on Tuesday December 8. Read pages 1 to 12 in
chapter 1 of Rudin. Do problems 1 to 5 on page 22, written up very neatly.
Also, do the following three problems:
1. Prove that in any ordered field, between any two distinct elements there
is at least one other element.
2. Prove that in any ordered field, between any two distinct element lie
in- finitely many elements.
3. Suppose that F is the set {0, 1} with the additive operations 0 + 0 =
0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0 and 0 · 0 = 0, 0 · 1 = 1 · 0 = 0, 1 · 1 = 1.
Prove that F is a field. Show that there is no order possible on F that
has the transitive property (x + y < x + z if y < z) and xy > 0 if x > 0
and y > 0.
General Advice on Writing the Proofs: Consider the following problem:
Suppose that x ∈ Q is non-zero and y ∈ R is irrational. Show that the
product xy is irrational.
Below is a good, not-so-good, and bad solution.
Good Solution: Suppose to the contrary that xy is rational. Since x is
non-zero we may divide by x to obtain
y=
xy
.
x
The right side above is a quotient of two rational numbers and hence rational, in contradiction to the hypothesis that y is irrational. The supposition
that xy is rational is thus false, so xy must be irrational.
Not-So-Good Solution: If xy rational, y = (xy)/x, y rational, contradiction.
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Bad Solution: The product xy must be irrational because if it were rational that would be impossible.
The good solution consists of complete, punctuated sentences. It also
makes use of all the hypotheses, including the essential assumption that x
is non-zero. The not-so-good solution omits this fact, and reads like it was
written by a caveman. The bad solution is completely lacking in insight and
merely asserts what is to be shown.
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