Question

Use the definition to prove that any denumerable set is
equinumerous with a

proper subset of itself. (This section is about infinite sets)

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.

41. Prove that a proper subset of a countable set is
countable

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

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

Is empty set a proper subset of a non-empty set? Why or why
not?

[Q] Prove or disprove:
a)every subset of an uncountable set is countable.
b)every subset of a countable set is countable.
c)every superset of a countable set is countable.

Prove that from any set A which contains 138 distinct integers,
there exists a subset B which contains at least 3 distinct integers
and the sum of the elements in B is divisible by 46. Show all your
steps

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

Prove that if a subset of a set of vectors is linearly
dependent, then the entire set is linearly dependent.

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...

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