Answers to Lianxi1
1.
Construct truth tables for the statements forms ((
p
→
q
)
∨ ¬
r
) and ((
p
→
(
q
→
r
))
→
((
p
→
q
)
→
(
p
→
r
))).
Solutions:
(1)((
p
→
q
)
∨¬
r
)
p
q
r
p
→
q
¬
r
(1)
0
0
0
1
1
1
0
0
1
1
0
1
0
1
0
1
1
1
0
1
1
1
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
1
1
1
1
1
1
1
0
1
(2) ((
p
→
(
q
→
r
))
→
((
p
→
q
)
→
(
p
→
r
)))
p
q
r
p
→
q
p
→
r
q
→
r
p
→
(
q
→
r
)
(
p
→
q
)
→
(
p
→
r
)
(2)
0
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
0
1
0
1
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
0
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
2.
Write the following sentences as statement forms,using statement letters to
stand for the atomic sentences,that is those sentences that are not built up out
of other sentences.
(a) If Mr Jones is happy, Mrs Jones is not happy,and if Mr Jones is not happy,
Mrs Jones is not happy.
(b) A suFcient condition for
x
to be odd is that
x
is prime.
(c) Either Sam will come to the party and Max will not, or Sam will not
come to the party and Max will enjoy himself.
(d) A necessary condition for a sequence
s
to converge is that
s
be bounded.
(e) If
a
is positive,
x
2
is positive.
Solutions:
(a) Let
p
:
Mr Jones is happy;
q
:
Mrs Jones is happy.
1
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(
p
→ ¬
q
)
∧
(
¬
p
→ ¬
q
)
(b) Let
p
:
x
is prime.
q
:
x
is odd.
Then
p
→
q
(c) Let
p
:
Sam will come to the party.
q
:
Max will come to the party.
r
:
Max will enjoy himself.
Then
(
p
∧¬
q
)
∨
(
¬
p
∧
r
)
(d) Let
p
:
A sequence
s
converges.
q
:
s
is bounded.
Then
p
→
q
(e) Let
p
:
x
is positive.
q
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 Fall '06
 Cheng
 Logic, Quantification, Mrs Jones, ¬b

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