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### Basic Truth Tables and Equivalents in Logic

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Date: 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.
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```
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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Associated Topics:
College Logic
High School Logic

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