Question

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. For those that are finite or uncountable, explain your reasoning.

a. integers that are divisible by 7 or divisible by 10

Answer #1

Which of the following sets are finite? countably infinite?
uncountable? Give reasons for your answers for each of the
following:
(a) {1\n :n ∈ Z\{0}};
(b)R\N;
(c){x ∈ N:|x−7|>|x|};
(d)2Z×3Z
Please answer questions in clear hand-writing and show me the
full process, thank you (Sometimes I get the answer which was
difficult to read).

(a) Let A and B be countably infinite sets. Decide whether the
following are true for all, some (but not all), or no such sets,
and give reasons for your answers. A ∪B is countably infinite A
∩B is countably infinite A\B is countably infinite, where A ∖ B =
{ x | x ∈ A ∧ X ∉ B }. (b) Let F be the set of all total unary
functions f : N → N...

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...

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 ≤...

For each of the following sets X and collections T of open
subsets decide whether the pair X, T satisfies the axioms of a
topological space. If it does, determine the connected components
of X. If it is not a topological space then exhibit one axiom that
fails.
(a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3},
{1, 2, 3, 4}}.
(b) X = {1, 2, 3, 4} and T...

Decide whether each of the given sets is a group with respect to
the indicated operation. If it is not a group, state a condition in
the deﬁnition of group that fails to hold.
(a) The set Z+ of all positive integers with operation
multiplication.
(b) For a ﬁxed integer n, the set of all complex numbers x such
that xn = 1 (That is, the set of all nth roots of 1), with
operation multiplication.
(c) The set Q'...

1. Determine in each of the following cases, whether the
described system is or not a group. Explain your answers. Determine
what of them is an Abelian group.
a) G = {set of integers}, a* b = a − b
b) G = {set of matrices of size 2 × 2}, A * B = A · B
c) G = {a0, a1, a2, a3,
a4}, ai * aj = a|i+j|,
if i+j < 5, ai *aj = a|i+j−5|,
if...

Let S = {0,2,4,6} and T = {1,3,5,7}. Determine
whether each of the following sets of ordered pairs is a function
with domain S and codomain T. If so, is it
one-to-one? Is it onto?
a. {(0,2),(2,4),(4,6),(6,0)}
b. {(6,3),(2,1),(0,3),(4,5)}
c. {(2,3),(4,7),(0,1),(6,5)}
d. {(2,1),(4,5),(6,3)}
e. {(6,1),(0,3),(4,1),(0,7),(2,5)}

Determine whether the data set is a population sample.Explain Your
reasoning.
1.The age of each State Governor
2.The speed of every fifth car passing a police trap
3.The annual salary for each employee at a company.
4.The number of pets in each U.S household.

For each of the
following data sets, choose the most appropriate response from the
choices below the table.
Data Set #1
Data Set #2
x
y
x
y
10
18
-22
92
13
12
-19
85
18
8
-16
71
24
2
-13
75
27
0
-10
68
32
4
-7
69
21
7
-5
66
41
13
0
62
45
17
2
59
a.
A strong positive linear relation exists
a.
A strong positive linear relation exists
b.
A...

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