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CSE 260
QUIZ5– Proofs
(20 minutes)
(30 points)
NAME:
1. (10 points) Use rules of inference (table given at the end) to show that
if
∀
x
(
P
(
x
)
∨
Q
(
x
)) and
∀
x
((
¬
P
(
x
)
∧
Q
(
x
))
→
R
(
x
)) are true,
then
∀
x
(
¬
R
(
x
))
→
P
(
x
)) is also true, where the domains of all quantiﬁers are the same.
Justify every step.
Step
Reason
1)
(
¬
P
(
a
)
∧
Q
(
a
))
→
R
(
a
)
by universal instantiation of the second premise.
2)
¬
(
¬
P
(
a
)
∧
Q
(
a
))
∨
R
(
a
)
by implication law from (1).
3)
(
P
(
a
)
∨ ¬
Q
(
a
))
∨
R
(
a
)
by De Morgan’s law from (2).
4)
(
R
(
a
)
∨
P
(
a
))
∨ ¬
Q
(
a
)
by associative and commutative laws from (3).
5)
P
(
a
)
∨
Q
(
a
)
by universal instantiation from the ﬁrst premise.
6)
(
R
(
a
)
∨
P
(
a
))
∨
P
(
a
))
by resolution from (4) and (5).
7)
R
(
a
)
∨
(
P
(
a
)
∨
P
(
a
))
by associative law from (6).
8)
R
(
a
)
∨
P
(
a
)
by idempotent law from (7).
9)
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This note was uploaded on 04/28/2008 for the course CSE 260 taught by Professor Saktipramanik during the Fall '08 term at Michigan State University.
 Fall '08
 SaktiPramanik

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