# GNED 1101 Chapter Notes - Chapter 1.4-1.5: Truth Table, Logical Biconditional, Contraposition

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22 May 2018
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1.4 Truth Tables for Conditional and Biconditional
Conditional Truth Table (If/Then)
P
Q
P Q
T
T
T
T
F
F
F
T
T
F
F
T
***A conditional is false only when the antecedent is true and the consequent is false
Example 1
o q p
P Q
Q
p
q p
T T
F
F
T
T F
T
F
F
F T
F
T
T
F F
T
T
T
Notice that pq, and q p have the same truth value in each of the four cases
o So whenever there is a conditional statement, you can reverse and negate the antecedent and
coseuet, ad the stateet’s tuth alue ill ot chage
o Ie; these two statements have the same truth value
If you’e cool, you ot ea clothig ith you school ae o it
If you wear clothing with your school name on it, youre not cool
Example 2 [(p q) p] q
P Q
P q
(false only when
both are false)
(pq) p
(
is true only
when p
q and
p
are true)
[(pq) p] q
(
is false only
when (p
q)
p)
is true and q is
false)
T T
T
F
T
T F
T
F
T
F T
T
T
T
T
F F
F
F
T
** a tautology is a compound statement that is always true, [(pq) p] q is a tautology
**conditional statements that are tautologies are called implications
Biconditional if and only if
The Biconditional statement pq means that pq and qp. This is written as
o (pq) (qp)
P Q
PQ
QP
(PQ) (QP)
T T
T
T
T
T F
F
T
F
F T
T
T
F
F F
T
T
T
A biconditional is true only when the component statements have the same truth value (both true or both
false)
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## Document Summary

***a conditional is false only when the antecedent is true and the consequent is false. Notice that p q, and q p have the same truth value in each of the four cases. So whenever there is a conditional statement, you can reverse and negate the antecedent and co(cid:374)se(cid:395)ue(cid:374)t, a(cid:374)d the state(cid:373)e(cid:374)t"s t(cid:396)uth (cid:448)alue (cid:449)ill (cid:374)ot cha(cid:374)ge. Ie; these two statements have the same truth value. If you"(cid:396)e cool, you (cid:449)o(cid:374)t (cid:449)ea(cid:396) clothi(cid:374)g (cid:449)ith you(cid:396) school (cid:374)a(cid:373)e o(cid:374) it. If you wear clothing with your school name on it, youre not cool. Example 2 [(p q) p] q. P q (false only when both are false) P (p q) p ( is true only when p q and p are true) [(p q) p] q ( is false only when (p q) p) is true and q is false) ** a tautology is a compound statement that is always true, [(p q) p] q is a tautology.

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