Download Problem Set 2

Problem Set # 2 / MATH 16200
Due: February 11, 2016 (Thursday) in class
1. Let f : X → Y be a function. Let I be an index set. Let {Aλ | λ ∈ I} be a collection of subsets of X,
and let {Bλ | λ ∈ I} be a collection of subsets of Y . Show the following.
!
[
[
(a) f
Aλ =
f (Aλ ).
λ∈I
λ∈I
!
\
(b) f
Aλ
⊂
λ∈I
\
f (Aλ ).
λ∈I
!
(c) f
−1
[
Bλ
=
λ∈I
[
f −1 (Bλ ).
λ∈I
!
(d) f −1
\
λ∈I
Bλ
=
\
f −1 (Bλ ).
λ∈I
(e) A ⊂ A0 ⇒ f (A) ⊂ f (A0 ).
(f) B ⊂ B 0 ⇒ f −1 (B) ⊂ f −1 (B 0 ).
(g) f −1 (B 0 \ B) = f −1 (B 0 ) \ f −1 (B).
(h) f (A0 \ A) ⊃ f (A0 ) \ f (A).
2. (a) Prove that ⊕ is commutative on R.
(b) Prove that ⊕ is associative on R.
(c) Use parts (a) and (b) to conclude that part (c) of Exercise 7.12 can safely be replaced by:
If A, B, C ∈ R, then A < B ⇐⇒ A ⊕ C < B ⊕ C.
(d) Prove that ⊗ is commutative on R.
(e) Prove that ⊗ is associative on R.
Note: The results of parts (a) and (b), (d), and (e) can be used in your proof of Theorem 7.22 from
Script #7.
3. Let Rn = {0, 1, . . . , n − 1} be the set of “integers modulo n” that we defined last quarter. For any
a ∈ Z, define the remainder modulo n of a to be the unique k ∈ Rn for which a − k is an integer that
is divisible by n. Define addition and multiplication on R as follows: a +n b is the remainder of a + b
modulo n, and a ·n b is the remainder of ab modulo n. Convince yourself (no need to turn this part in)
that all field axioms, except possibly FA8, are satisfied in Rn .
(a) When is Rn a field? (That is, when does Rn satisfy FA8?) Prove your answer.
(b) Can Rn ever be an ordered field? Prove your answer.
4. Define A := {x ∈ Q | 1 ≤ x ≤ 2} and define f : A → Q by f (q) := q 2 . In the following paragraph,
there is a lie (in fact, there are several of them, but one principal lie upon which all of the other lies
depend). In the following paragraph, find the principal lie and explain what is wrong with the reasoning.
By inspection, f (1) = 1 and f (2) = 4. Since 2 is between 1 and 4, by Theorem 5.17 (the Intermediate Value Theorem), there must be some q ∈ A such that f (q) = 2. Therefore, for this q, q 2 = 2,
and hence, there is, in fact, a rational number whose square is 2.
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5. Consider the field Q. Denote by F the collection of all functions Q → Q. We define addition and
multiplication operations on F as follows. If f, g ∈ F , then
(f + g) (x) := f (x) + g (x)
and
(f g) (x) := f (x) g (x) .
In other words, f + g is the map x 7→ f (x) + g (x) and f g is the map x 7→ f (x) g (x).
(a) Prove that F is not a field (despite the fact that it has been labelled by the letter “F ”).
(b) Among the ten field axioms, which of them does F satisfy?
(c) Theorems 6.13 and 6.14 are often called, respectively, the “additive cancellation law” and the
“multiplicative cancellation law”. Prove that the additive cancellation law holds in F , but the
multiplicative cancellation law does not hold in F .
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