Question

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

Answer #1

Prove directly (using only the definition of the countably
infinite set, without the use of any theo-rems) that the union of a
finite set and a countably infinite set is countably infinite.

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!

Determine whether each of these sets is finite, countably
infinite, or uncountable. For those that are countably infinite,
exhibit a one-to-one correspondence between the set of positive
integers and that set. For those that are finite or uncountable,
explain your reasoning.
a. integers that are divisible by 7 or divisible by 10

Prove : If S is an infinite set then it has a subset A which is
not equal to S, but such that A ∼ S.

Prove that if X and Y are disjoint countably infinite sets then
X ∪ Y is countably infinity (can you please show the bijection from
N->XUY clearly)

Prove Cantor’s original result: for any nonempty set (whether
finite or infinite), the cardinality of S is strictly less than
that of its power set 2S . First show that there is a one-to-one
(but not necessarily onto) map g from S to its power set. Next
assume that there is a one-to-one and onto function f and show that
this assumption leads to a contradiction by defining a new subset
of S that cannot possibly be the image of...

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
Z
×{
0
,
1
,
2
,
3
}
is countably infinite by finding a bijection
f
:
Z
→
Z
×{
0
,
1
,
2
,
3
}
. (
Hint:
Consider the
division algorithm

(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...

Suppose that E is a closed connected infinite subset of a metric
space X. Prove that E is a perfect set.

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