28 ~ Theory of Computer Science
Opposite-If the triangle is not isoceles, then two of its sides are not
equal.
Contrapositive-If two of the sides of a triangle are not equal, then
.the triangle is not isoceles.
(b) Converse-If the Indians won't go to the USA for employment, then
there is no unemployment in India.
Opposite-If there is unemployment in India. then the Indians will go
to the USA for employment.
(c) Contrapositive-If the Indians go to the USA for employment, then
there is unemployment in India.
EXAMPLE 1.29
Show that:
(-, P /\ (-, Q /\ R)) v (Q /\ R) v (P /\ R) ¢:::> R
Solution
(-, P /\ (-, Q /\ R) v (Q /\ R) v (P /\ R)
¢:::> «-, P /\ -, Q) /\ R) v (Q /\ R) v (P /\ R) by using the associative law
¢:::> (-, (P v Q) /\ R) v (Q /\ R) v (P /\ R) by using the DeMorgan's law
¢:::> h (P v Q) /\ R) v (Q v P) /\ R)
by using the distributive law
¢:::> (-, (P v Q) v (P v Q) /\ R
by using the commutative
and distributive laws
by using Is
by using 1 9
EXAMPLE 1.30
Using identities, prove that:
Q v (P /\ -, Q) V (-, P /\ -, Q) is a tautology
Solution
Q v (P /\ -, Q) V (-, P /\ -, Q)
¢:::> «Q v P) /\ (Q V ---, Q) v (-, P 1\ -, Q) by using the distributive law
¢:::> «Q v P) /\ T) v (-, P /\ -, Q)
by using Is
¢:::> (Q v P) v ---, (P v Q)
by using the DeMorgan' slaw
and 1 9
¢:::> (P V Q) V -, (P v Q)
by using the commutative
law
¢:::>T
Hence the given fonnula is a tautology.
by using Is
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