Question

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!

Answer #1

a) Let A and B be two countably infinite sets. This implies that there exists bijections . Then, let us look at the function , where

Note that if x is in A, then all of the even numbers are covered and if x is in B\A, then all the odd numebrs will be covered. Hence, . Now, as f and g are bijective, hence h is injective. Thus h is a bijection therefore is a countably infinite set.

b) The result would still hold if we take countable number of countably infinte sets, i.e. the cardinality would be the same as as if you keep doing unions of the sets, the cardinality remains the same due to the two set case in part a).

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

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) 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...

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 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.

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 for each of the following:
a. Exercise A union of finitely many or countably many countable
sets is countable. (Hint: Similar)
b. Theorem: (Cantor 1874, 1891) R is uncountable.
c. Theorem: We write |R| = c the “continuum”. Then c = |P(N)| =
2א0
d. Prove the set I of irrational number is uncountable. (Hint:
Contradiction.)

Which of the following sets are finite? countably infinite?
uncountable? Give reasons for your answers for each of the
following:
(a) {1\n :n ∈ Z\{0}};
(b)R\N;
(c){x ∈ N:|x−7|>|x|};
(d)2Z×3Z
Please answer questions in clear hand-writing and show me the
full process, thank you (Sometimes I get the answer which was
difficult to read).

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...

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