MAD 2104
May 6, 2010
Quiz I and Key
Prof. S. Hudson
1) [25pts] Verify the absorption law using a truth table (and a brief explanation):
p
∨
(
p
∧
q
)
≡
p
.
2) [25pts] Find a compound proposition in disjunctive normal form that is true when
exactly one of
p,q
and
r
is true (and is false otherwise). Hint: we did this one in class.
3) [50pt] Answer each part with “True” or “False”. You do not have to justify your answers.
a)
∀
x
∈
R,
(
x
2
≥
x
).
b)
∀
x
(
P
(
x
)
∧
Q
(
x
))
≡ ∀
x,P
(
x
)
∧∀
x,Q
(
x
).
c)
¬
p
→
(
p
→
q
) is a tautology.
d)
∀
x,
∃
y,P
(
x,y
)
≡ ∃
y,
∀
x,P
(
x,y
).
e)
¬
(
∀
x,
∃
y,P
(
x,y
))
≡ ∃
x,
∀
y,
¬
P
(
x,y
).
Remarks and Answers:
The average among the top 20 students was 82 / 100. Here is
a rough scale for the quiz, based mainly on that average:
As 87 to 100
Bs 77 to 86
Cs 67 to 76
Ds 57 to 66
1) Draw a truth table with 4 rows and approx 4 columns, labeled
p
,
q
,
p
∧
q
and
p
∨
(
p
∧
q
)
(there is some ﬂexibility here). Point out that the truth values under
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