590 Appendix A • Logic and Proofs- A necessary condition for an integer to be divisible by 12 is that it be
divisible by both 4 and 3. - A sufficient condition for an integer to be divisible by 12 is that it be
divisible by both 4 and 3.
12. It isn't true that if you go on diet X you will get sick and not lose weight.
PART B:- Suppose you know that PA rv Q, Q V R, and R =? S are all true. Find
the only possible truth-values of P, Q, R, and S.
14. Suppose you know that P V R , (P V Q) =? S, (RAP)=? T, and rv Sare
all true. Find the only possible truth-values of P, Q, R , S, and T.- Let A denote "The assignment is easy," W denote "I work hard,'' U denote
"I understand the basic ideas," and E denote "I enjoy mathematics."
Translate the following statements into symbolic form:
(a) The assignment is easy only if I understand the basic ideas.
(b) Enjoying mathematics is a necessary condition for me to work
hard.
(c) The assignment isn't easy if I don't work hard or do not under-
stand the basic ideas.
(d) The assignment is easy only if I work hard, but it is not true
that my enjoying mathematics is a sufficient condition for my un-
derstanding the basic ideas.Translate the following symbolic forms into verbal statements:(e) (UAW)=? E.
(f) rv A:::} (rv UV rv W).
(g) rv (E =? (W =?A)).LOGICALLY EQUIVALENT PROPOSITIONS
Definition A.1.14 A tautology is a compound proposition that always has
truth-value T, regardless of the truth-values of its constituent parts.Examples A.1.15 Some obvious tautologies:
(a) P V rv P (d) (PA Q) :::} Q
(b) P =? P (e) rv (PA rv P) (law of excluded middle)
(c) P =? (P v Q) (f) rv (rv P) {::} P (double negation)