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Truth TablesTruth Tables
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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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