Question

· 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 countably infinite.

Answer #1

(a) Let A and B be countably infinite sets. Decide whether the
following are true for all, some (but not all), or no such sets,
and give reasons for your answers. A ∪B is countably infinite A
∩B is countably infinite A\B is countably infinite, where A ∖ B =
{ x | x ∈ A ∧ X ∉ B }. (b) Let F be the set of all total unary
functions f : N → N...

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

Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers
in the interval (0, 1). (Hint: Prove this by contradiction.
Specifically, (i) assume that there are countably infinite real
numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii)
express each real number x1 between 0 and 1 in decimal expansion;
(iii) construct a number y whose digits are either 1 or 2. Can you
find a way...

Prove the union of two infinite 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.

a) Prove that the union between two countably infinite sets is a
countably infinite set.
b) Would the statement above hold if we instead started with an
infinite amount of countably infinite sets?
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Thank you in advance!

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...

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.”

If A and B are both uncountably infinite sets then A - B could
be? Select one of the following options:
a) Uncountably infinite
b) Finite
c) Countably infinite
Explain the solution.

Prove that tue union of countable sets is countable.

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