Question

9. Let S = {a,b,c,d,e,f,g,h,i,j}.

a. is {{a}, {b, c}, {e, g}, {h, i, j}} a partition of S?
Explain.

b. is {{a, b}, {c, d}, {e, f}, {g, h}, {h, i, j}} a partition
of S? Explain. c. is {{a, b}, {c, d}, {e, f}, {g, h}, {i, j}} a
partition of S? Explain.

Answer #1

A set P is a partition of a set S if and only if :

1) The union of all the elements of P is equal to S.

2) The intersection of any two distinct elements of P always comes out to null.

That is, if we partition set S into three parts that is , then and .

**a)** This is not a partition of S. Since, d and f
are missing, the union of the elements will not be equal to S.
[Point 1 is violated]

**b)** This is not a partition. Since h is being
repeated in two distinct elements, the intersection of the elements
will not be null. [Point 2 is violated].

**c)** Yes, this is a partition of S. Since the
union of all distinct elements will be equal to S itself and the
intersection will be null.

Given that, for some a, b, c, d, e, f, g, h, i ∈ R, [a b c d e f
g h i ] = 5, evaluate the following determinants:
(c) [ka ld mg
kb le mh
kc lf mi] Here, k, l, and m are non-negative constants.

Using this matrix.
A =
a
b
c
d
e
f
g
h
i
Suppose that det(A) = 5. Find the determinant of the
following matrix.
B =
a + 3g
b + 3h
c + 3i
-g
-h
-i
4d
4e
4f

let
A = { a, b, c, d , e, f, g} B = { d, e , f , g}
and C ={ a, b, c, d}
find :
(B n C)’
B’
B n C
(B U C) ‘

Consider the following bivariate data.
Point
A
B
C
D
E
F
G
H
I
J
x
9
8
9
3
7
3
3
4
6
5
y
6
2
2
3
6
3
3
2
4
6
(a) Construct a scatter diagram of the given bivariate data. (Do
this on paper. Your instructor may ask you to turn in this
work.)
(b) Calculate the covariance. (Give your answer correct to two
decimal places.)
(c) Calculate sx and sy.
(Give...

Consider the following bivariate data.
Point
A
B
C
D
E
F
G
H
I
J
x
0
1
1
2
3
4
5
6
6
7
y
5
5
8
3
4
1
2
0
1
1
(a) Construct a scatter diagram of the given bivariate data. (Do
this on paper. Your instructor may ask you to turn in this
work.)
(b) Calculate the covariance. (Give your answer correct to two
decimal places.)
(c) Calculate sx and sy.
(Give...

Consider the following bivariate data.
Point
A
B
C
D
E
F
G
H
I
J
x
4
5
2
6
7
6
2
5
6
5
y
2
6
4
7
7
0
0
1
5
2
(a) Construct a scatter diagram of the given bivariate data. (Do
this on paper. Your instructor may ask you to turn in this
work.)
(b) Calculate the covariance. (Give your answer correct to two
decimal places.)
(c) Calculate sx and sy.
(Give...

Let G and H be groups and f:G--->H be a surjective
homomorphism. Let J be a subgroup of H and define f^-1(J) ={x is an
element of G| f(x) is an element of J}
a. Show ker(f)⊂f^-1(J) and ker(f) is a normal subgroup of
f^-1(J)
b. Let p: f^-1(J) --> J be defined by p(x) = f(x). Show p is
a surjective homomorphism
c. Show the set kef(f) and ker(p) are equal
d. Show J is isomorphic to f^-1(J)/ker(f)

Let S = {a, b, c, d, e, f} with P(b) = 0.21, P(c) = 0.11, P(d) =
0.11, P(e) = 0.18, and P(f) = 0.19. Let E = {b, c, f} and F = {b,
d, e, f}. Find P(a), P(E), and P(F).

Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many diﬀerent functions are there from S to
T?
b. How many diﬀerent one-to-one functions are there from S to
T?
c. How many diﬀerent one-to-one functions are there from T to
S?
d. How many diﬀerent onto functions are there from T to
S?

Suppose a 3x3 real matrix A=[a b c; d e f; g h i] has
determinant 5. What is the determinant of the 3x3 matrix B=[2a 2c
2b; 2d 2f 2e; 2g 2i 2h]?

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