Homework 6 due:
April, 26, 2013, 8:00am — exam time
The due time is strict. No late homework will be accepted, as I am leaving for a conference shortly
after the semester is over.
Problems marked with ∗ will be graded. Please practice on the rest of them to the extent you find
useful.
1. ∗Show that
∣{1, 2, 3, 4}∣ = ∣{3, 4, 5, 6}∣
2. ∗Show that
∣{a ∈ Z ∶ [a] = [1] in Z3 }∣ = ∣Z∣
3. Prove from the definition of multiplication of cardinal numbers that
κ ⋅ (λ ⋅ µ) = (κ ⋅ λ) ⋅ µ
4. ∗Prove from the definition of addition and multiplication of cardinal numbers that
κ ⋅ (λ + µ) = κ ⋅ λ + κ ⋅ µ
5. Prove from the definition of addition, multiplication and exponentiation of cardinal numbers that
κλ+µ = κλ ⋅ κµ
6. ∗Let c denote the cardinality of real numbers, c = ∣R∣. What is cℵ0 ? In other words what is the
cardinality of a set of all real valued sequences? Hint: use properties Cop1)-Cop16).
7. Simplify ℵ0 ⋅ (c + c)
8. ∗Let κ be any cardinal number κ ≥ 2. Can κκ = κ?
9. ∗Prove that multiplication on finite cardinals numbers matches with multiplication on non-negative
integers, in other words, prove that for all n, m ∈ Z, n, m ≥ 0
∣{1, 2, . . . , n}∣ ⋅ ∣{1, 2, . . . , m}∣ = ∣{1, 2, . . . , n ⋅ m}∣
Hint: Consider any n, prove by induction on m, using the fact that
∣{1, 2, . . . , k}∣ + ∣{1, 2, . . . , r}∣ = ∣{1, 2, . . . , k + r}∣
for all k, r ∈ Z, k, r ≥ 0
10. Consider a summation of non-negative real numbers ax > 0 over an set X of any cardinality (possibly
uncountable). It is formally defined as a supremum of the sums over finite subsets of X:
∑ ax = sup { ∑ ax ∶ A ⊂ X, A is finite}
x∈X
x∈A
Where a sum over a finite set ∑x∈A ax is defined in a standard way. You do not have to understand
this definition to the full extent, it is enough to use the fact that
B ⊂ X ⇒ ∑ ax ≤ ∑ ax
x∈B
x∈X
.
Prove that if ∑x∈X ax < ∞ then the set {x ∈ X ∶ ax > 0} is countable, so in fact the summation is
over a countable set, so we could rewrite it as ∑∞
n=1 axn .
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Hint: Prove first that for each m > 0 a set {x ∈ X ∶ ax > m
} is finite, and then use the fact that a
countable sum of finite sets is countable, which is a consequence of the fact that ℵ0 ⋅ ℵ0 = ℵ0
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