(3), (4), Modus Ponens,
C(b)
(6)
(5), (6), Modus Ponens,
Y(b)
(7)
By con junc tion,
C b Y b
( )
( )
∧
Conclusion: It is good for the country and for you to buy an expensive car.
Ì Exam ple 8.4.9: Determine the validity of the argument: “It is not the
case that some trigonometric functions are not periodic. Some periodic
functions are continuous. Therefore, it is not true that all trigonometric
functions are not continuous”.
Construct proof, if it is valid.
Solu tion
T(x): x is a Trigonometric function.
P(x): x is a periodic function
C(x): x is a continuous function.
¬ ℑ
→ ¬
[
( )
( )]
x T x
P x
∨ ¬
→
T x
P x
( )
( )
ℑ
→
x P x
C x
( )
( )
ℑ
→
x P x
C x
( )
( )
∴¬ ∨
→ ¬
(
( )
( )
x T x
C x
∴ ℑ ¬
→
(
( )
( ))
x T x
C x
1. ∨ ¬
→
T x
P x
( )
( )
P
2.
¬
→
T a
P a
( )
( )
US, (1)
3.
ℑ
→
x P x
C x
( )
( )
4.
P a
C a
( )
( )
→
(3), ES
5.
¬
→
T a
C a
( )
( )
(2), (4), Impli ca tion
6.
ℑ ¬
→
x
T x
C x
( )
( )
(5), EG
⇒ Asser tion is VALID.
Ì Exam ple 8.4.10: Show that ( )( ) ( , ) ( )( ) ( , )
∨ ∨
→ ¬ ∨
x
y
y
P x y
x
P x y is
logically valid.
Solu tion
1.
( ) ( ) ( , )
∨ ∨
x
y P x y
P
2.
( ) ( , )
∨ y P a y
US, (1)
3.
P a y
( , )
UG, (2)
4.
( ) ( , )
ℑ x P a y
EG, (3)
5.
( )
( , )
ℑ ∨
x
y P a y
UG, (4)
288
Theory of Automata, Formal Languages and Computation