21-12∞/15-151 Infinite sets review problems
Clive Newstead, Friday 11th November 2016
Questions marked ? are particularly challenging.
1. Let X be a set. Define what it means to say that X is finite, and define what it means to
say that X is countably infinite.
2. Let X and Y be sets. For each of the following, say (without proof) whether it is true or
false. If the statement is false, provide a counterexample.
(a) If there is an injection X → C for some countable set C, then X is countable.
(b) If there is an surjection X → C for some countable set C, then X is countable.
(c) If there is an injection C → X for some countable set C, then X is countable.
(d) If there is an surjection C → X for some countable set C, then X is countable.
(e) If X and Y are countable, then X × Y is countable.
(f) If X × Y is countable, then X is countable and Y is countable.
(g) If X and Y are countable, then X ∪ Y is countable.
(h) If X ∪ Y Is countable, then X is countable and Y is countable.
(i) If X and Y are countable, then the set Y X of functions X → Y is countable.
(j) If X ⊆ Y and Y is countable, then X is countable.
3. (a) Find a surjection f : N × N → Z.
(b) Deduce that Z countably infinite.
(c) Find a surjection g : Z × (Z \ {0}) → Q.
(d) Deduce that Q is countably infinite.
4. Prove that the set of all binary sequences of even length is countable.
5. ? A function f : N → N is finitely supported if the set {n ∈ N | f (n) 6= 0} is finite. Prove
that the set of all finitely supported functions N → N is countable.
6. ? A sequence (an )n∈N of real numbers is weakly decreasing if an ≥ an+1 for all n ∈ N.
Prove that the set of weakly decreasing sequences of natural numbers is countably infinite.
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7. Given a set Σ, thought of as an alphabet, let Σ∗ denote the set of finite strings from Σ,
thought of as a set of words. (For example, {0, 1}∗ is the set of finite binary strings.)
(a) Suppose f : Σ → N is an injection, and let k ∈ N. Prove that the function g : Σk → N
defined by
f (s )
g(s1 , s2 , . . . , sk ) = 2f (s1 ) · 3f (s2 ) · · · · · pk k
is injective, where pk is the k th -smallest positive prime.
(b) Deduce that the set of words from a countable alphabet is countable.
(c) Deduce that the set of finite sequences of natural numbers is finite.
8. For each of the following, say whether it is true or false, and prove your claim.
(a) There are countably many finite subsets of Z.
(b) There are countably many infinite subsets of Z.
9. Suppose {Un | n ∈ N} is a partition of R. Can each set Un in the partition be countable?
10. Prove that the set NN of all functions N → N is uncountable.
11. Let X be a set. For each of the following, say whether it is true or false; if the statement is
false, provide a counterexample.
(a) If there is an injection X → C for some uncountable set C, then X is uncountable.
(b) If there is an surjection X → C for some uncountable set C, then X is uncountable.
(c) If there is an injection C → X for some uncountable set C, then X is uncountable.
(d) If there is an surjection C → X for some uncountable set C, then X is uncountable.
12. Prove that the set of weakly decreasing sequences of integers is uncountable.
13. For each of the following, say whether it is true or false, and prove your claim.
(a) If U ⊆ R is uncountable then R \ U is countable.
(b) If U ⊆ R is countable then R \ U is uncountable.
(c) Every open interval (a, b) ⊆ R with a < b is uncountably infinite.
14. ? A set X ⊆ P(N) has the property that, for all A, B ∈ X, either A ⊆ B or B ⊆ A. Can X
be uncountable?
15. ? A line in R2 is a subset ` ⊆ R2 of the form
` = {(a + tb, c + td) | t ∈ R}
for fixed a, b, c, d ∈ R. Prove that R2 is not a countable unionSof lines; that is, there does
not exist a countable set {`n | n ∈ N} of lines in R2 such that n∈N `n = R2 .
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