(a) P P
P
| ⇔ ¬
(To prove)
We know P Q
P Q
P P
P P
P
|
(
)
|
(
)
.
⇔ ¬ ∧
⇔ ¬ ∧
⇔ ¬
Hence proved.
(b) ( | ) | ( | )
P P Q Q
P Q
⇔ ∨
(To prove)
We have from (a), P P
P
Q Q
Q
P P Q Q
P Q
P
Q
P Q
|
|
( | ) | ( | )
|
(
)
.
⇔ ¬
⇔ ¬
⇔ ¬ ¬
⇔ ¬ ¬ ∧ ¬
⇔ ∨
Hence proved.
(c) ( | ) | ( | )
P Q P Q
P Q
⇔ ∧
(To prove)
P Q
P Q
|
(
)
= ¬ ∧
( | ) | ( | )
[ (
)]|[ (
)]
[[ (
) [ (
))]
P Q P Q
P Q
P Q
P Q
P Q
⇔ ¬ ∧
¬ ∧
⇔ ¬ ¬ ∧ ∧ ¬ ∧
⇔ ¬
∨ ∧ ∨
⇔ ¬ ∨
⇔ ∧
[(
) (
)
(
)
.
P Q
P Q
P Q
P Q
Hence proved.
Ì Exam ple 8.1.21: Establish the following implications:
(a) ¬ ⇒
⇒
P
P Q
(
)
(b) ¬ ⇒
⇒
(
)
P Q
P
(c) ¬ ∧
⇒
⇒ ¬
Q P Q
P
(
)
Solu tion
(a) ¬ ⇒
⇒
P
P Q
(
)
P
Q
¬ P
P Q
⇒
¬ ⇒
⇒
P
P Q
(
)
T
T
F
T
T
T
F
F
F
T
F
T
T
T
T
F
F
T
T
T
∴ It is a Tautology.
Hence proved.
Prop o si tions and Pred i cates
263