It can be shown that R (the set of all real numbers) is an infinite-dimensional vectorspace over Q (field of rationals).
Is it true that any basis (by basis I mean algebraic basis or Hamel basis) of R over Q has to be uncountable ?

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Yes. It will have to have the same cardinality as R, in fact.
Let B be a basis of R over the field Q.
We know that every element of R can be ...

Given a set of all real numbers containing 4s and 5s in their decimal
representation. Determine the given set is countable or uncountable.
Example : numbers in this set: 4.4455544, 455.55554444
You don't need to prove this, but rather explain it to me why.
I understand why from 0-1 is uncountable or from 1-2 etc using

Please see the attached image for questions I and II.
III) If A is a countable subset of an uncountable set X, prove that X A is uncountable.
IV) Suppose that f is a function from X into Y so that the range of f is uncountable. Prove that X is uncountable.
V) Prove that the set of all polynomials with rational coeffici

Answer the following by establishing 1-1(one to one) correspondence with a set of known cardinality:
1 - Is the set of all functions from{0,1} to N countable or noncountable?
2 - Is the set of all functions from N to {0,1} countable or noncountable?
3 - Given a set B ,a subset A of P(B) is called an antichain if no element of

Prove that there exists a subjection from P(N) onto omega_1, where N is the set of all natural numbers, P(N) is the power set of N, and omega_1 is the least uncountable ordinal.
See attachments for fully formulated problem.

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

Please help with the following problem.
A) Let P be the set of all functions f : N -----> N such that for some M in N and all n>M, f(n + 1) = f (n). In other words, f is in P provided that after some point, f is constant. Show that P is countable.
b) Let E be the set of all strictly increasing functions f : N ----> N. Show

For any set B, let P(B) denote the power set of B (the collection of all subsets of B):
P(B) = {E: E is a subset of B}
Let A be a countably infinite set (an infinite set which is countable), and do the following:
(a) Prove that there is a one-to-one correspondence between P(A) and the set S of all countably infinite seq

Consider the compound statement (P ^ Q) V (~ P ^ R)
a) Find the truth table for the statement.
b) IS the statement a tautology?
c) In the following program code, what has to be the output for the answr to be "yes"?
...
2. Given the following sets A = {1.2,3.4}, B = {2,3,4,5}, and C= {2,4,6}
a) Find ....
4. Prove that