Question

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

Answer #1

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

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

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

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...

[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 the set of odd numbers is infinite.

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

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

Prove that The set P of all prime numbers is a closed subset of
R but not an open subset of R.

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