Question

[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.

Answer #1

I gave the counter example which is not true

true or false?
every uncountable set has a countable subset. explain

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

why
is every countable subset a zero set? real analysis

Prove or disprove:
If A and B are subsets of a universal set U such that A is not a
subset of B and B is not a subset of A, then A complement is not a
subset of B complement and B complement is not a subset of A
complement

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.]

Verify: any countable ordered set is similar to a subset of Q
intersect (0,1).

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

Let f : A → B and g : B → C. For each of the statements in this
problem determine if the statement is true or false. No explanation
is required. Just put a T or F to the left of each statement.
a. g ◦ f : A → C
b. If g ◦ f is onto C, then g is onto C.
c. If g ◦ f is 1-1, then g is 1-1.
d. Every subset of...

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

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.

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