7. Write down each of the following statements in the form “if p, then q” in
English.
(a) Your guarantee is good, only if you bought your PC less than 1
year ago.
(b) If you drive more than 800 kilometres, you have to buy diesel.
8. State the converse and contrapositive of each of the following
implications.
(a) A positive integer is a prime only if it has no divisors other than 1
and itself.
(b) I go to class whenever there is a test.
9. Construct a truth table for each of the following compound propositions.
(a) (
)
p q r
∨ ∨
(b) (
)
p q r
∧ ∧
(c) (
)
p q
r
∧ ∨ ¬
10. Construct a Truth table for each of the following compound
propositions.
(a) (
) (
)
p q
p r
→ ∨ ¬ →
(b) (
) (
)
p q
p r
→ ∧ ¬ →
(c) ¬ → →
p
q r
(
)
11. Show that ¬ ∨
(
)
p q and ¬ ∧ ¬
p
q are logically equivalent by
constructing Truth Table.
12. Show that the propositions p q
→ and ¬ ∨
p q are logically equivalent.
13. Show that the propositions p q r
∨ ∧
(
) and (
) (
)
p q
p r
∨ ∧ ∨
are
logically equivalent.
14. Show that ¬ ∨ ¬ ∧
(
(
))
p
p q and ¬ ∧ ¬
p
q are logically equivalent.
15. Show that p
p
≡ ¬ ¬
(
).
16. Rewrite the statement: “It is not true that I am not happy” in simpler
form.
17. Show that ¬ ∧ ≡ ¬ ∨ ¬
(
) (
) (
)
p q
p
q (De Morgan’s Laws)
18. Given p: The president is a Democrat
q: The president is a Republican
Express (a) ¬ ∧
(
)
p q (b) (
) (
)
¬ ∨ ¬
p
q .
19. Show that the statement p
p
∨ ¬
(
) is a tautology.
20. Show that (
) [(
) (
)]
p q
p
q
∨ ∨ ¬ ∧ ¬
is a tautology.
21. Show that the statement (
) [(
) (
)]
p q
p
q
∨ ∧ ¬ ∧ ¬
is a contradiction.
22. Simplify the statement ¬
∧ ¬
∧
([
(
)] ).
p
q
r
23. Consider: “You will get an A if either you are clever and the sun shines,
or you are clever and it rains”. Rephrase the conditions more simply.
24. Show that p q
p q
→ ≡ ¬ ∨
(
)
using Truth Table construction.
25. Verify the “Switcheroo Law” using Truth Table. p q
p q
→ ≡ ¬ ∨
(
)
.
Prop o si tions and Pred i cates
295