is numerically equivalent to [0,1].

Timothy Miles
Math 352-01
Section 6.2
1, 12
1
a) Prove that every closed interval [a, b] is numerically equivalent to [0, 1].
Numerical equivalence requires a bijection:
Define f : R → R
Given a < b, f(a) = 0, f(b) = 1 a linear formula with slope = (1 − 0)/(b − a)
f(a) = 1 = a 1
−a
+ y-intercept = a/(b − a)+ y-intercept ⇒ y-intercept =
b−a b−a
f : f(x) = x 1 − a
b−a b−a
Surjective: Let set A represent the interval [a, b] and B represent the interval [0, 1]
If x = a, f(a) = 0 ∈ B and f(b) = 1 ∈ B
Since this is a linear formula with slope 1 and (b − a) > 0 slope is positive and constant.
b−a
Therefore by Intermediate value theorem ∀x ∈ A, f(x) ∈ B
Injective: Let f(x) = f(y) Show x = y
f(x) = x 1 − a
1
and f(y) = y − a ⇒
b−a b−a b−a b−a
x 1 − a = y 1 − a ⇒ x 1 = y 1 ⇒ x = y ∴
b−a b−a b−a b−a b−a b−a
b) Prove that any two closed intervals [a, b] and [c, d] are numerically equivalent.
As ealier, need to define a linear formula s.t.
f : R → R
Given a < b, f(a) = c, f(b) = d with slope d−c
b−a
Then f(x) = x d−c
and y-intercept = c − a
d−c
b−a
d−c
+ (c − a )
b−a b−a
Surjective: Let set A represent the interval [a, b] and B represent the interval [c, d]
If x = a, f(a) = c ∈ B and f(b) = d ∈ B
Since this is a linear formula with slope d−c and (b − a) > 0, (d − c) > 0 slope is positive
b−a
and constant. Therefore by Intermediate value theorem ∀x ∈ A, f(x) ∈ B
Injective: Let f(x) = f(y) Show x = y
x d−c
d−c
+ (c − a ) = y d−c
d−c
+ (c − a ) ⇒ x d−c = y d−c ⇒ x(d − c) = y(d − c) ⇒ x = y ∴
b−a b−a b−a b−a b−a b−a
12 Let A, B be sets contained in a universal set U such that A ⊆ B. Suppose that A is
countable and B is uncountable. Prove that B − A is uncountable.
Proof by contradiction: Let B − A be countable
Then B ∩ A ⊆ B by Theorem 2.2.1 Part 6
Also the subset of a countable set is also countable
1

We assumed B − A to be countable however, B − A is only countable when B is countable,
but we were given that it wasn’t
Therefore given A ⊆ B ∩ A, A countable and B uncountable B − A is uncountable
2