Question

Prove for each:

a. Proposition: If A is finite and B is countable, then A ∪ B is countable.

b. Proposition: Every subset A ⊆ N is finite or countable. [Similarly if A ⊆ B with B countable.]

c. Proposition: If N → A is a surjection, then A is finite or countable. [Or if countable B → A surjection.]

Answer #1

[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 the union of a finite collection of countable
sets is countable.

Proposition 16.4 Let S be a non–empty finite set.
(a) There is a unique n 2 N1 such that there is a 1–1
correspondence from {1, 2,...,n} to S.
We write |S| = n. Also, we write |;| = 0.
(b) If B is a set and f : B ! S is a 1–1 correspondence, then B is
finite and |B| = |S|.
(c) If T is a proper subset of S, then T is finite and |T| <...

Prove that the set of all finite subsets of Q is countable

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

Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...

2. Prove that the set of finite sequences of lower case letters
(a, b, c, . . ., z) is countable.

Prove whether or not the set ? is countable.
a. ? = [0, 0.001)
b. ? = ℚ x ℚ
I do not really understand how to prove
S is countable.

Prove: Proposition 11.13. Congruence modulo n is an equivalence
relation on Z :
(1) For every a ∈ Z, a = a mod n.
(2) If a = b mod n then b = a mod n.
(3) If a = b mod n and b = c mod n, then a = c mod n

Is the set of all finite subsets of N countable or uncountable?
Give a proof of your assertion.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 16 minutes ago

asked 16 minutes ago

asked 21 minutes ago

asked 29 minutes ago

asked 45 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago