M42 — Discrete Mathematics
Spring 2011
M. Stanley
Sample First Exam
Do not use any aids (calculators, computers, notes, etc.) on this exam. When asked to provide proofs, be
sure to give careful, detailed and complete explanations in full English sentences. Form is as important as
content.
(1)
(
1 point each
)
Let
A
=
{
1
,
2
}
and
B
=
{
2
,
3
}
. Let the universal set
U
=
{
1
,
2
,
3
,
4
}
. Find
(a)
A
∪
B
=
{
1
,
2
,
3
}
(b)
A
∩
B
=
{
2
}
(c)
A
⊕
B
=
{
1
,
3
}
(d)
A
×
B
=
{
(1
,
2)
,
(1
,
3)
,
(2
,
2)
,
(2
,
3)
}
(e)
(
A
) =
{∅
,
{
1
}
,
{
2
}
,
{
1
,
2
}}
(f)
A
\
B
=
{
1
}
=
{
2
,
3
,
4
}
(2)
(
3 points
)
Use a truth table to show that ((
p
∨ ¬
q
)
→
q
)
≡
q
.
Answer:
p
q
¬
q
p
∨ ¬
q
(
p
∨ ¬
q
)
→
q
T
T
F
T
T
T
F
T
T
F
F
T
F
F
T
F
F
T
T
F
Because the pattern of
T
’s and
F
’s in the
q
column and the (
p
∨ ¬
q
)
→
q
column match, these two formulas
are logically equivalent.
(3)
(
3 points
)
If possible, find an assignment of
T
and
F
to the propositional variables
p
,
q
,
r
, and
s
such
that (
p
→
q
)
→
(
r
→
s
) is
T
, but (
p
→
r
)
→
(
q
→
s
) is
F
.
p
F
q
T
r
F
s
F