# HW02 - CSE260 Solutions to Homework Set#2 6 Use a truth...

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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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HW02 - CSE260 Solutions to Homework Set#2 6 Use a truth...

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