Basic Truth Tables and Equivalents in LogicDate: 05/23/2000 at 16:45:11 From: Carissa Subject: Truth tables What are the truth tables for p and q, including p^q, p<->q, p<-q, and p->q? I'm studying for my regents exam and I need to brush up on this area. Date: 05/24/2000 at 12:22:15 From: Doctor TWE Subject: Re: Truth tables Hi Carissa - thanks for writing to Dr. Math. Here are the basic truth tables and some useful equivalents, I'm assuming that you know how to use them in logic problems. IMPLICATION (p implies q) EQUIVALENCE NEGATION CONJUNCTION DISJUNCTION (if p then q) (p equiv. q) (not p) (p and q) (p or q) (p only if q) (p iff q) p | ~p p q | pq p q | pvq p q | p->q p q | p=q -------- ---------- ----------- ------------ ----------- T | F T T | T T T | T T T | T T T | T F | T T F | F T F | T T F | F T F | F F T | F F T | T F T | T F T | F F F | F F F | F F F | T F F | T Note that there are different symbols used for these operations, depending on the system used. Here are some of the more common alternatives: Negation: Tilde (~) before the proposition, bar above the proposition. Conjunction: No symbol between propositions, carat (^) or 'cap' between propositions, dot between propositions. (Also related to intersection, usually represented by an inverted 'U'.) Disjunction: 'v' or 'cup' between propositions, plus sign (+) between propositions. (Also related to union, usually represented by a 'U'.) Implication: Right arrow (->) between propositions, 'U' turned 90 degrees counterclockwise between propositions. (Sometimes these are written "backwards"; q<-p is logically equivalent to p->q. The former is usually read 'q if p' or 'q is implied by p'.) Equivalence: Equivalence symbol (similar to the equals sign (=), but with 3 parallel segments) between propositions, double arrow (<->) between propositions. (Though they are not technically the same, frequently the equals sign (=) is used in place of the equivalence symbol when typing on a computer, because the latter is not part of most standard keyboard symbol sets.) Another logic connective that you might run into is the exclusive-or. This is more commonly used in electronic logic circuits than in propositional calculus (the study of logic). Its truth table is the opposite of the equivalence truth table (i.e. the outputs are F T T F when the tables are written as above). Symbols used for exclusive-or include a circled plus sign, an equivalence sign with a slash (/) through it (read 'p not equivalent to q'), or sometimes a circled 'v'. Here are some useful equivalencies: (Note that T is any tautology - a proposition that's always true, and F is a contradiction - a proposition that's always false.) p->q = ~pvq Def. of implication p<->q = (p->q)(q->p) Def. of equivalence p(+)q = ~pq v p~q Def. of ex-or ~(~p) = p Double negation ~T = F ~F = T Def. of negations pvq = qvp pq = qp Commutative laws (pvq)vr = pv(qvr) p(qr) = (pq)r Associative laws p(qvr) = pq v pr pv(qr) = (pvq)(pvr) Distributive laws pvF = p pT = p Identity laws pv~p = T p(~p) = F Negation laws pvp = p pp = p Idempotent laws ~(pvq) = (~p)(~q) ~(pq) = ~pv~q DeMorgan's laws pvT = T pF = F Universal bound laws pv(pq) = p p(pvq) = p Absorption laws pv(~pq) = pvq p(~pvq) = pq " " I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/ |
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