Question

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

Answer #1

A countable union of disjoint countable sets is countable
Note: countable sets can be either finite or infinite
A countable union of countable sets is countable
Note: countable sets can be either finite or infinite

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 that tue union of countable sets is countable.

Prove the union of two infinite countable sets is countable.

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.”

SHow that a union of a finite or countable number of
sets of lebesgue measure zero is a set of lebesgue measure
zero.
Please show all steps

Prove that the union of infinitely many sets is countable using
induction.

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.]

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

Prove that the union of two compact sets is compact using the
fact that every open cover has a finite subcover.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 16 minutes ago

asked 23 minutes ago

asked 32 minutes ago

asked 46 minutes ago

asked 46 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago