Solu tion
P
P
Q Q
R
P
P
Q
Q R
P
P
Q
Q
∨ ¬ →
∨ → ¬
⇔ ∨ ¬ →
∨ ¬ ∨
⇔ ∨ ¬ →
∨ ¬ ∨
(
(
(
)))
(
(
(
)))
(
[(
) (
)]
(
[
(
)]
(
(
))
[ (
) (
)
Q R
P
P
T Q R
P
P
Q R
P
P
Q R
∨
⇔ ∨ ¬ → ∨ ∨
⇔ ∨ ¬ →
∨
⇔ ∨ ¬ ¬ ∨ ∨ ]
[
(
)]
(
) [
(
)]
(
)
⇔ ∨ ∨ ∨
⇔
∨ ∨ ∨ ∨
⇔ ∨ ∨
⇔ ∨ ∨ ∨
⇔ ∨ ∨
P P Q R
P P
P Q R
P Q R
P P Q R
P Q R.
(Answer)
Ì Exam ple 8.5.2: Obtain the disjunctive Normal form of
P
P
Q R
∨ ¬ ∧ ¬ ∨
(
)
Solu tion
P
P
Q R
P
P
P
Q
P R
P P
Q
P R
Q
∨ ¬ ∧ ¬ ∨
⇔
∨ ¬ ∧ ∨ ¬ ∨ ∨
⇔ ∧ ∨ ¬ ∨ ∨
⇔ ¬ ∨
(
)
(
) (
) (
)
(
) (
)
(
)
P R
P
Q R
∨
⇔ ∨ ¬ ∨
(Answer).
Ì Exam ple 8.5.3: Obtain the disjunctive normal form of
(
) (
)
¬ ∨ ¬ → ¬ ∧
P
Q
P R
Solution:
(
)
(
)
[[ (
)] (
)]
[(
) (
)]
¬ ∨ ¬ → ¬ ∧
⇔ ¬ ¬ ∨ ¬
∨ ¬ ∧
⇔
∧ ∨ ¬ ∧
P
Q
P R
P
Q
P R
P Q
P R
(Answer).
Ì Exam ple 8.5.4: Obtain the disjunctive normal form of
(
(
)) (
)
P
Q R
P Q
∧ ¬ ∧
∨ →
Solu tion
(
(
)) (
)
(
(
)) (
)
(
(
)) (
P
Q R
P Q
P
Q R
P Q
P
Q R
P
∧ ¬ ∧
∨ →
⇔
∧ ¬ ∧
∨ ¬ ∨
⇔
∧ ¬ ∨
∨ ¬ ∨ Q
P
Q
P R
P Q
P
Q
P
P
Q Q
P R
)
(
) (
) (
)
(
) (
) (
)
⇔
∧ ¬ ∨ ∧ ∨ ¬ ∨
⇔
∧ ¬ ∨ ¬ ∨ ∧ ¬ ∨ ∨ ∧
Prop o si tions and Pred i cates
291