Question

Show that the set A = {1, 2, 3, 4, …} is equinumerous with the set B = {3, 6, 9, 12, …} by constructing a bijection from A to B. (Prove that it is a bijection)

Answer #1

1) Given set A and {$} where {$} represents set with only one
element. Prove there is bijection between A x {$} and A.
2) Given sets A, B. Prove A x B is equivalent to B x A using
bijection.
3) Given sets A, B, C. Prove (A x B) x C is equvilaent to A x (B
x C) using a bijection.

3. Let T be the set of integers that are not divisible
by 3. Prove that T is a countable set by finding a bijection
between the set T and the set of integers Z, which we know is
countable from class. (You need to prove that your function is a
bijection.)

Let E = {0, 2, 4, . . .} be the set of non-negative even
integers
Prove that |Z| = |E| by defining an explicit bijection

let the universal set be U = {1, 2, 3, 4, 5, 6, 7, 8, 9} with A
= {1, 2, 3, 5, 7} and B = {3, 4, 6, 7, 8, 9}
a.)Find (A ∩ B) C ∪ B
b.) Find Ac ∪ B.

4. Let set U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20}
set A = numbers in U that divide into 12 with no remainder,
set B = numbers in U that divide into 16 with no remainder, and set
C = the numbers in U that divide into 20 with no remainder.
a. Made a Venn diagram showing the elements of the sets U, A,...

One number is randomly selected from the following set: { 1, 2,
3, 4, 5, 6, 7, 8, 9, 10 }.
Let
A = event
that the selected number is even
B = event that the selected number is a multiple of 3
Find the following
probabilities.
a) P( A
and B
b) P( A
or B
)
c) P(
A B)
d) Are events A...

1) a) From the set {-8, -2/3, 5i, √(-9), √2, 0, 3+3i, -2.35,
7}
i) List the set of Natural Numbers
ii) List the set of Integers
iii) List of the set of Rational Numbers
vi) List the set of Real Numbers
2)Solve the following pairs of simultaneous equations 3x + y = 7
and 2x - 2y = 2
b) i) -30 ÷ -6 - (-12 + 8) – 4 x 3 =
c)Calculate the simple interest earned if...

Consider these data sets:
I: 1, 3, 2, 2, 5, 4, 4, 3, 3
II: 1, 2, 4, 1, 2, 5, 2, 5, 1, 5, 5, 3
(a) Find the mean and median for each set. Show your work.
(b) Find variance and standard deviation for each set. Show your
work.
(c) Would you be surprised to hear someone claim that these data
were drawn from the same population? (Hint: Draw a histogram for
each set. Compare the shapes.)

Show (-1,1)~R (where R= set of real numbers) by f(x)=
x/(1-|x|)
Use this to show g(x)=x/(d-|x|) is also a bijection (i.e. g:
(-d,d)->R)
Finally consider h(x)= x + (a+b)/2 and show it is a bijection
where h: (-d,d)->(a,b)
Conclude: R~(a,b)

For the following set of data,
X
Y
1
0
2
2
3
4
4
6
5
8
a. Compute the Pearson correlation, r, for this data set.
b. Re-arrange the Y scores so that the value of r = –1.00.

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