Question

Prove that the set of all finite subsets of Q is countable

Answer #1

Is the set of all finite subsets of N countable or uncountable?
Give a proof of your assertion.

Prove the union of a finite collection of countable
sets is countable.

Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]

Let S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

[Q] Prove or disprove:
a)every subset of an uncountable set is countable.
b)every subset of a countable set is countable.
c)every superset of a countable set is countable.

For a set A, let P(A) be the set of all subsets of A. Prove that
A is not equivalent to P(A)

Use the fact that “countable union of disjoint countable sets is
countable" to prove “the set of all polynomials with rational
coefficients must be countable.”

41. Prove that a proper subset of a countable set is
countable

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

6. Let S be a finite set and let P(S) denote the set of all
subsets of S. Define a relation on P(S) by declaring that two
subsets A and B are related if A ⊆ B.
(a) Is this relation reflexive? Explain your
reasoning.
(b) Is this relation symmetric? Explain your
reasoning.
(c) Is this relation transitive? Explain your
reasoning.

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