Question

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.]

Answer #1

[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.

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

Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...

Prove whether or not the set ? is countable.
a. ? = [0, 0.001)
b. ? = ℚ x ℚ
I do not really understand how to prove
S is countable.

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

Prove: Proposition 11.13. Congruence modulo n is an equivalence
relation on Z :
(1) For every a ∈ Z, a = a mod n.
(2) If a = b mod n then b = a mod n.
(3) If a = b mod n and b = c mod n, then a = c mod n

Prove that a subset of a countably infinite set is finite or
countably infinite.

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...

Assume that X and Y are finite sets. Prove the following
statement:
If there is a bijection f:X→Y then|X|=|Y|.
Hint: Show that if f : X → Y is a surjection then |X| ≥ |Y| and if
f : X → Y is an injection then
|X| ≤ |Y |.

If A and B are two finite sets. Prove that |A ∪ B| = |A| + |B| −
|A ∩ B| is true.

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