Truth Tables Presented by: Tutorial Services The Math Center.

Truth TablesTruth Tables

Presented by:Presented by:

Tutorial ServicesTutorial Services

The Math CenterThe Math Center

Truth TablesTruth Tables

A truth table is a device used to determine A truth table is a device used to determine when a compound statement is true or false.when a compound statement is true or false.

Formal Formal NameName

SymbolSymbol ReadRead Symbolic Symbolic FormForm

NegationNegation ~~ ““Not”Not” ~p~pConjunctionConjunction ““And”And” p qp qDisjunctionDisjunction ““Or”Or” p qp qConditionalConditional ““If-then”If-then” p q p q

Bi-conditionalBi-conditional ““If and only If”If and only If” p q p q

Connectives used in truth tables:

Types of ArgumentsTypes of Arguments

pp qq

TT TT TT

TT FF FF

FF TT FF

FF FF FF

qp Conjunction

•When finding the truth value of a conjunction, all values must be true in order for the entire conjunction to be true.

•For example, if p and q are true, then (p q) is true.

•For example, if p is true and q is false, (p q) is false.

•For example, if p and q are false, then (p q) is false.

Types of ArgumentsTypes of Arguments

pp qq

TT TT TT

TT FF TT

FF TT TT

FF FF FF

qp Disjunction

•When finding the truth value of a disjunction, only one value needs to be true in order for the entire disjunction to be true.

•For example, if p is true and q is false, then (p q) is true.

•For example, if both p and q are true, then (p q) is true.

•For example, if both p and q are false, then (p q) is false.

Types of ArgumentsTypes of Arguments

pp ~p~p

TT FF

FF TT

Negation

•The truth values of ~p are exactly the opposite truth values of p.

•For example, true for p would be false for ~p.

•For example, false for (p q) would be true for ~(p q).

Types of ArgumentsTypes of Arguments

pp qq p qp q

TT TT TT

TT FF FF

FF TT TT

FF FF TT

Conditional

•When finding the truth value of a conditional statement, same values will be true. Otherwise, follow the truth value of the conclusion (which is the second proposition).

•For example, if p and q are false, then (p q) is true.

•For example, if p is true and q is false, then (p q) is false.

•For example, if p is false and q is true, then (p q) is true.

Types of ArgumentsTypes of Arguments

pp qq

TT TT TT

TT FF FF

FF TT FF

FF FF TT

Bi-conditional

qp •When finding the truth value of a bi-conditional statement, same values will be true. Otherwise, the truth value will be false.

•For example, if both p and q are false, then (p q) is true.

•For example, if p is true and q is false, then (p q) is false.

•For example, if p is false and q is true, then (p q) is false.

ExamplesExamples

pp qq p p q q ~q~q

TT TT TT FF TT

TT FF FF TT TT

FF TT FF FF FF

FF FF FF TT FF

Example1:

qqp ~)(

qqp ~)(

ExamplesExamples

pp qq rr ~r~r q ~rq ~r

TT TT TT FF FF FF

TT TT FF TT TT TT

FF FF TT FF FF TT

FF TT TT FF FF TT

TT FF FF TT FF FF

TT FF TT FF FF FF

FF TT FF TT TT TT

FF FF FF TT FF TT

Example 2:

)~( rqp )~( rqp

ExamplesExamples

pp qq rr ~r~r q ~rq ~r

TT TT TT FF FF FF

TT TT FF TT TT TT

FF FF TT FF FF TT

FF TT TT FF FF TT

TT FF FF TT FF FF

TT FF TT FF FF FF

FF TT FF TT TT FF

FF FF FF TT FF TT

Example 3:

)~( rqp )~( rqp

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