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Logic: Truth Tables Constructing a Truth Table

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Truth Table A truth table for a compound statement is a list of the truth or falsity of the statement for every possible combination of truth and falsity of its components. In other words, a truth table helps to show whether a statement is true or false.

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Rows To find the number of rows used in a truth table, take the number 2 raised to the power of the number of variables. For example, if there was a p statement and a q statement, there would be 2 variables, 2^2 is 4. If there were three statements, it would be 2^3, or 8 rows.

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Columns The columns under the connectives /\, and \/, stand for the conjunction, and disjunction of the expression on the two sides of that connective.

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Two statement table pq T Half the rows should be true T The rows should alternate T, F TF The result should be F Half the rows should be false T one of every possibility FFTT, TF, FT, FF

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The three statement table pqr T Half of the rows T Alternate TT T Alternate T should be true, T and FF F T and F T the other half FT so that there is T should be false. FF one of every FTT possibility FTF TTT, TTF, TFT FFT ect. FFF

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Negation Truth Table p~ p The opposite of p is ~p TFNot true is false FTNot false is true

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Conjunction Truth Table pq p /\ qp and q TTT True only if both are true. TFF FTF FFF

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Disjunction Truth Table pqp \/ qp or q TTT True if either on is true TFT FTT FFF False only if both are false

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Lets fill out a table pqp\/ (or) (~p (not) /\ (and) q) TT TF FT FF

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Negate the p column pqp\/ (or) (~p (not) /\ (and) q) TTF TFF FTT FFT

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Copy the q column pqp\/ (or) (~p (not) /\ (and) q) TTFT TFFF FTTT FFTF

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Fill the /\ column pqp\/ (or) (~p (not) /\ (and) q) TTFFT TFFFF FTTTT FFTFF

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Copy the p column pqp\/ (or) (~p (not) /\ (and) TTTFF TFTFF FTFTT FFFTF

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Fill in the \/ column using the p and the /\ columns pqP\/ (or) P and (~p/\p) (~p (not) /\ (and) (~p) and (p) p TTTTFFT TFTTFFT FTFTTTF FFFFTFF

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Use the final column to determine what type of statements it is \/ (or) P and (~p/\p) Tautology Always True Contradiction Always False Contingency Sometimes true, sometimes false Txx Txx Txx Fxx

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Contingency Some were true, while one was false. That makes this statement a contingency.

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Real life example In case that was not entirely clear, lets take a look at an everyday example. Circuits. There are two different kinds of circuits, a series circuit and a parallel circuit. When the switch is closed the light will be on. However, with a series circuit, both switches have to be closed and with a parallel circuit only one switch has to be closed for the light to go on.

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Series Circuits Switch pSwitch qLight Closed On Only on if both are closed ClosedOpenOff OpenClosedOff Open Off

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Parallel Circuits Switch pSwitch qLight Closed On ClosedOpenOn OpenClosedOn Open Off Only off when both are open

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Conclusion That concludes the Logic: Truth Tables lesson. For more information, consult Finite Mathematics by Berresford and Rockett. Or learn logic online: 2.htm 2.htm

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