MATH 3012 Sample Quiz Questions for Test 1, Spring 2006 WTT
Note 1:
There are approximately two to three times as many problems listed here as you can
expect on an hour exam, but this more comprehensive version should be of greater assistance to
students in studying for the test.
Note 2
:
Often professors give tests with instructions like:
Be sure to explain your answers.
1.
How many 14-letter words can be formed using the 26 letters of the alphabet if:
a. Repetition of letters is allowed.
Answer:
26
14
.
b. Repetition of letters is not allowed.
Answer:
P
(26
,
14).
c. Each word contains exactly 4 vowels, with repetition of letters allowed.
Answer:
(
14
4
)
5
4
21
10
.
d. Each word contains exactly 4 vowels, with repetition of letters not allowed.
Answer:
(
14
4
)
P
(5
,
4)
P
(21
,
10).
2.
Let
A
=
{
1
,
2
,
3
,
4
,
5
}
and
B
=
{
a, b, c, d, e
}
. Determine if the following relations are functions.
If they are functions, determine if they are surjections, injections or bijections.
a.
R
1
=
{
(1
, a
)
,
(2
, b
)
,
(3
, d
)
,
(4
, c
)
,
(5
, e
)
}
.
Answer:
R
1
is a function. It is a bijection.
b.
R
2
=
{
(1
, a
)
,
(2
, b
)
,
(3
, c
)
}
.
Answer:
R
2
is not a function since there is no element
y
∈
B
for
which (4
, y
)
∈
R
2
, i.e., there is no image of the element 4
∈
A
.
c.
R
3
=
{
(1
, a
)
,
(2
, b
)
,
(2
, c
)
,
(3
, d
)
,
(4
, e
)
}
.
Answer:
R
3
is not a function since it contains the
pairs (2
, b
) and (2
, c
). To be a function from
A
to
B
, it is necessary that for each
a
∈
A
, there
is only one
y
∈
B
for which (
a, y
) belongs to the relation.
3.
Let
X
=
{
1
,
2
,
3
,
4
,
5
}
. Which of the following relations are equivalence relations on
X
? Which
are partial orders on
X
?
a.
R
1
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
3)
,
(4
,
4)
,
(5
,
5)
}
.
Answer:
R
1
is both a partial order and an equiva-
lence relation. Note, as a partial order,
R
1
determines an antichain.
b.
R
2
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
3)
,
(4
,
4)
,
(5
,
5)
,
(2
,
4)
,
(4
,
3)
,
(2
,
5)
}
.
Answer:
R
2
is neither a partial or-
der nor an equivalence relation. It violates the transitive requirement. Note that (2
,
4)
,
(4
,
3)
∈
R
2
but (2
,
3)
∈
R
2
.
c.
R
3
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
3)
,
(4
,
4)
,
(5
,
5)
,
(2
,
4)
,
(4
,
3)
,
(2
,
3)
,
(2
,
5)
}
Answer:
This relation is a
partial order but not an equivalence relation.
d.
R
4
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
3)
,
(4
,
4)
,
(5
,
5)
,
(2
,
4)
,
(4
,
2)
,
(5
,
3)
,
(3
,
5)
}
.
Answer:
This relation is
an equivalence relation but not a partial order.
e.
R
5
=
{
(1
,
1)
,
(2
,
2)
,
(3
,
3)
,
(4
,
4)
,
(5
,
5)
,
(2
,
4)
,
(4
,
2)
,
(5
,
3)
,
(3
,
5)
,
(2
,
3)
}
Answer:
This relation
is neither.
4.
KK Bakery sells 8 varieties of donuts. Coffee can be ordered black, with milk, with sugar or
with both.