Question

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.

Answer #1

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

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.

Use the definition to prove that any denumerable set is
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proper subset of itself. (This section is about infinite sets)

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...

Prove the statements (a) and (b) using a set element proof and
using only the definitions of the set operations (set equality,
subset, intersection, union, complement):
(a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A.
(b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B.
(c) Now prove the statement from part (b)

Prove directly from definition that the set {1/2^n | n=1, 2, 3,
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Theorem 16.11 Let A be a set. The set A is infinite if and only
if there is a proper subset B of A
for which there exists a 1–1 correspondence f : A -> B.
Complete the proof of Theorem 16.11 as follows: Begin by
assuming that A is infinite.
Let a1, a2,... be an infinite sequence of distinct elements of A.
(How do we know such a sequence
exists?) Prove that there is a 1–1 correspondence between the...

Prove the product rule for derivatives using only the following
definition of derivative:
[f(x) - f(a)] / (x-a)

Prove using the definition of truth that for any first-order
formula φ, φ is valid iff ∀x(φ) is valid.

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