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**Constructing a Truth Table**

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 p q T Half the rows should be true**

The rows should alternate T, F F The result should be Half the rows should be false one of every possibility TT, TF, FT, FF

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**The three statement table**

p q r T Half of the rows Alternate TT Alternate should be true, and FF F T and F the other half so that there is should be false. one of every possibility TTT, TTF, TFT ect.

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**Negation Truth Table p ~ p The opposite of p is ~p T F**

“Not true” is “false” “Not false” is “true”

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**Conjunction Truth Table**

p q p /\ q p and q T True only if both are true. F

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**Disjunction Truth Table**

p q p \/ q p or q T True if either on is true F False only if both are false

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Lets fill out a table p q \/ (or) (~p (not) /\ (and) q) T F

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Negate the p column p q \/ (or) (~p (not) /\ (and) q) T F

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Copy the q column p q \/ (or) (~p (not) /\ (and) q) T F

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Fill the /\ column p q \/ (or) (~p (not) /\ (and) q) T F

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Copy the p column p q \/ (or) (~p (not) /\ (and) T F

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**Fill in the \/ column using the p and the /\ columns**

q P \/ (or) P and (~p/\p) (~p (not) /\ (and) (~p) and (p) T F

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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 T x F

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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, let’s 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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**Only on if both are closed**

Series Circuits Switch p Switch q Light Closed On Only on if both are closed Open Off

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**Only off when both are open**

Parallel Circuits Switch p Switch q Light Closed On 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:

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2012 Pearson Education, Inc. Slide 3-2-1 Chapter 3 Introduction to Logic.

2012 Pearson Education, Inc. Slide 3-2-1 Chapter 3 Introduction to Logic.

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