CSE260
Solutions to Homework Set #2
6.
Use a truth table to verify the first De Morgan law ¬(p /\ q) ≡ ¬p \/ ¬q.
We see that the fourth and seventh columns are identical.
p
q
p /\ q
¬(p /\ q)
¬p
¬q
¬p \/ ¬q
T
T
T
F
F
F
F
T
F
F
T
F
T
T
F
T
F
T
T
F
T
F
F
F
T
T
T
T
24.
Show that (p → q) \/ (p → r) and p → (q \/ r) are logically equivalent.
We determine exactly which rows of the truth table will have T as their entries.
Now (p → q) \/ (p →
r) will be true when either of the conditional statements is true.
The conditional statement will be true
if p is false or if q in one case or r in the other case is true, i.e., when q \/ r is true, which is precisely
when p → (q \/ r) is true.
Since the two propositions are true in exactly the same situations, they are
logically equivalent.
(p → q) \/ (p → r)
≡ (¬p \/ q) \/ (¬p \/ r)
(Implication Laws)
≡ ¬p \/ q \/ ¬p \/ r
(Associative Laws)
≡ ¬p \/ ¬p \/ q \/ r
(Commutative Laws)
≡ ¬p \/ q \/ r
(Idempotent Laws)
≡ p → q \/ r
(Implication Laws)
44.
Show that ¬ and /\ form a functionally complete collection of logical operators.
Given a compound proposition p, we can, by Exercise 43, write down a proposition q that is logically
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 Spring '08
 SaktiPramanik
 Logic, Logical connective

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