Question

Show that the following property holds for countable sets:

if S_1, S_2,S_3,... is a sequence of countable sets of real numbers, then the set S formed by taking all the elements that belong to at least one of the sets S_i is also a countable set.

Answer #1

Let B1, B2, B3, . . . be a sequence of sets, each of which is
countable. Prove that the union, A = S∞ n=1 Bn, of all of the sets
in the sequence is a countable set.

SHow that a union of a finite or countable number of
sets of lebesgue measure zero is a set of lebesgue measure
zero.
Please show all steps

Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers
in the interval (0, 1). (Hint: Prove this by contradiction.
Specifically, (i) assume that there are countably infinite real
numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii)
express each real number x1 between 0 and 1 in decimal expansion;
(iii) construct a number y whose digits are either 1 or 2. Can you
find a way...

1.
Let A and B be sets. The set B is of at least the same size as
the set A if and only if (mark all correct answers)
there is a bijection from A to B
there is a one-to-one function from A to B
there is a one-to-one function from B to A
there is an onto function from B to A
A is a proper subset of B
2.
Which of these sets are countable? (mark all...

For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning). For countable
sets, provide some explicit counting scheme and list the first 20
elements according to your scheme. (a) The set [0, 1]R ×
[0, 1]R = {(x, y) | x, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
(b) The set [0, 1]Q × [0, 1]Q = {(x, y) |
x, y ∈ Q, 0 ≤ x ≤...

(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...

11. (6 Pts.) Show that if we split any 11 numbers in 5 sets,
then there exists one set that contains a subset such
that the sum of its elements is a multiple of 3.

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

Let (sn) be a sequence. Consider the set X consisting of real
numbers x∈R having the following property: There exists N∈N s.t.
for all n > N, sn< x. Prove that limsupsn= infX.

15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
cardinality as all nonnegative real numbers. (Hint: What is a
function whose graph goes from 0 to positive infinity as x goes
from 0 to pi/2?)
c) Use parts a and b to...

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