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

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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
consequent, and the statement’s truth value will not change
o Ie; these two statements have the same truth value
If you’re cool, you wont wear clothing with your school name on it
If you wear clothing with your school name on it, youre not cool
Example 2 [(p q) p] q
P Q
p
(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
F
F
T
T F
F
F
T
F T
T
T
T
F F
T
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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