Ì Exam ple 8.1.4: Given
p : It is cold;
q : 12 + 5 = 200
and r : It rains.
Express the following in symbolic form.
(a) It is cold only if 12 + 5 = 200.
(b) A necessary condition for it to be cold is that 12 + 5 = 200.
(c) A sufficient condition for it to be cold is that 12 + 5 = 200.
(d) It rains and 12 + 5 is not 200.
(e) It never rains when 12 + 5 = 200.
Solu tion
(a) p q
⇔
(b) p q
⇔
(c) q
p
⇔
(d) r
p
∧ ¬
(e) q
r
⇒ ¬
Ì Exam ple 8.1.5: Let
p : prices are high,
q : wages are increasing.
Express the following in verbal form.
(a) p q
∧
(b) ¬ ∧ ¬
p
q
(c) ¬ ∧
(
)
p q
(d) p
q
∨ ¬
(e) ¬ ¬ ∨ ¬
(
)
p
q .
Solu tion
(a) p q
∧ : Prices are high and wages are increasing
(b) ¬ ∧ ¬
p
q: Prices are not high and wages are not increasing
(c) ¬ ∧
(
)
p q : It is not that prices are high and wages are increasing
(d) p
q
∨ ¬ : Prices are high or the wages are not increasing
(e) ¬ ¬ ∨ ¬
(
)
p
q : It is not that prices are not high or wages are not
increasing.
Ì Exam ple 8.1.6: Determine the truth of the following:
(a) 5 < 6 and 6 is a positive integer.
(b) 5 > 6 or 6 is a positive integer
(c) If 5 > 6, then 100 is a prime number
(d) If 10 > 6, then 100 is a prime number.
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Theory of Automata, Formal Languages and Computation