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

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

coseuet, ad the stateet’s tuth alue ill ot chage

o Ie; these two statements have the same truth value

If you’e cool, you ot ea clothig ith you school ae 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)

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

T

F

F

T

T F

T

F

F

T

F T

T

T

T

T

F 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.