Question

- 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

Answer #1

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.”

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

Prove the union of two infinite countable sets is 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 tue union of countable sets is countable.

Suppose A is an infinite set and B is countable and disjoint
from A. Prove that the union A U B is equivalent to A by defining a
bijection f: A ----> A U B.
Thus, adding a countably infinite set to an infinite set does
not increase its size.

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...

Using the following theorem: If A and B are disjoint denumerable
sets, then A ∪ B is denumerable, prove the union of a finite
pairwise disjoint family of denumerable sets {Ai
:1,2,3,....,n} is denumerable

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

· Let A and B be sets. If A and B are countable, then A ∪ B is
countable.
· Let A and B be sets. If A and B are infinite, then A ∪ B is
infinite.
· Let A and B be sets. If A and B are countably infinite, then A
∪ B is countably infinite.
Find nontrivial sets A and B such that A ∪ B = Z, then use these
theorems to show Z is...

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