Question

1. Let T = {(1, 2), (1, 3), (2, 5), (3, 6), (4, 7)}. T : X -> Y. X = {1, 2, 3, 4}, Y = {1, 2, 3, 4, 5, 6, 7}

a) Explain why T is or is not a function.

b) What is the domain of T?

c) What is the range of T?

d) Explain why T is or is not one-to one?

Answer #1

Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an
example of a function f: A -> A with the indicated properties,
or explain why no such function exists.
(a) f is bijective, but is not the identity function f(x) =
x.
(b) f is neither one-to-one nor onto.
(c) f is one-to-one, but not onto.
(d) f is onto, but not one-to-one.

x
7
0
3
4
3
1
y
6
2
6
5
6
7
Assuming that the regression equation is y = 4.433 + 0.300x and
that the SSE = 12.6333, test to determine if the slope is not equal
to zero using alpha = 0.10
a) Test Statistic T = ______ (round to two decimal places)
b) Critical value(s) = _______, _______ (round to 3 decimal
places)
c) P-value = ______

Let X = {1, 2, 3, 4} and Y = {2, 3, 4, 5}. Define f : X → Y by
1, 2, 3, 4 → 4, 2, 5, 3. Check that f is one to one and onto and
find the inverse function f -1.

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 5 5 2 4 4 6 3 3 7 5 3 6 3 4 4 6 5 4 5 3 7 5 5 4 2 6 5 6 6]
dataset.
As you can see Y is a discrete variable. Please write down a
probability mass function for Y. Remember the example of pmf for
die rolling experiment;

The domain of the function g(x) is -1<x<6 and
the range is -5<y<10.
Find the domain and range of the following given functions.
a). the domain of y=g(x-4) is?
b.) the range of y=g(x)+2 is?

4. Let A = {1, 2, 3, 4, 5}. Let L = {(x, y) ∈ A × A : x < y}
and B = {(x, y) ∈ A × A : |x − y| = 1}.
i. Draw graphs representing L and B.
ii. Determine L ◦ B.
iii. Determine B ◦ B.
iv. Is B transitive? Explain.

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

Let G = 〈(1 2 3 4 5 6), (1 6)(2 5)(3 4)〉. Let H1 :=
〈(1 4)(2 5)(3 6)〉 and H2 := 〈(1 6)(2 5)(3 4)〉. Determine
if the subgroups H1 and H2 are normal
subgroups of G.

LINEAR ALGEBRA
For the matrix B=
1 -4 7 -5
0 1 -4 3
2 -6 6 -4
Find all x in R^4 that are mapped into the zero vector by the
transformation Bx.
Does the vector:
1
0
2
belong to the range of T? If it does, what is the pre-image of
this vector?

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