MAD 2104
Sept 1, 2011
Quiz I
Prof. S. Hudson
1) Prove that
¬
p
↔
q
is logically equivalent to
p
↔ ¬
q
using a table and a few words (you
can use another proof style, for partial credit, at your own risk).
2a) Find the negation of
∀
x
(
x
2
> x
) (answer without using
¬
).
2b) The proposition in 2a) is true for some domains and not others. Give an example of
a domain for which it is true. Your answer should be a set of numbers.
2c) Give a domain for which it is false.
3) Answer each part with “True” or “False”. You don’t have to explain (but it doesn’t
hurt, and might help if we decide later that a question was not totally clear).
a)
p
∨
(
p
→
q
) is a tautology.
b) (
p
→
q
)
∧
(
p
→ ¬
q
) is a contradiction.
c) The converse of
p
→ ¬
q
is
q
→ ¬
p
.
d)
p
→ ¬
q
is logically equivalent to
q
→ ¬
p
.
e) 0101
L
1100 = 1101.
Bonus (approx 5 pts): Show that
{∧
,
¬}
is functionally complete. You can quote any HW
problems you did related to this. Hint: Find an equivalence that shows
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 Fall '08
 STAFF
 Logic, partial credit, Prof. S. Hudson, typical good answer

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