Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.2 Truth Tables for Negation,...

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 3.2

Truth Tables for Negation,

Conjunction, and

Disjunction


What You Will Learn

Truth tables for negations,

conjunctions, and disjunctions

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Truth Table

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

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Negation Truth Table

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p ~pCase 1 T FCase 2 F T


Compound Statement Truth Table

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p qCase 1 T TCase 2 T FCase 3 F TCase 4 F F


Conjunction Truth Table

The conjunction is true only when both p and q are true.

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p q p ⋀ qCase 1 T T TCase 2 T F FCase 3 F T FCase 4 F F F


Disjunction Truth Table

The disjunction is true when either p is true, q is true, or both p and q are true.

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p q p ⋁ qCase 1 T T TCase 2 T F TCase 3 F T TCase 4 F F F


Negation

Negation ~p is read “not p.”

If p is true, then ~p is false;if p is false, then ~p is true.In other words, ~p will always have the opposite truth value of p.

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Conjunction

Conjunction p ⋀ q is read “p and q.”

p ⋀ q is true only when both p and q are true

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Disjunction

Disjunction p ⋁ q is read “p or q.”

p ⋁ q is true when either p is true or q is true, or both p and q are true.

In other words, p ⋁ q is false only when both p and q are false.

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Constructing Truth Tables1. Determine if the statement is a negation, conjunction, disjunction, conditional, or biconditional.The answer to the truth table appears under:

~ if it is a negation⋀ if it is a conjunction ⋁ if it is a disjunction

→ if it is conditional↔ if it is biconditional

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Constructing Truth Tables

2. Complete the columns under the simple statements, p, q, r, and their negations ~p, ~q, ~r, within parentheses, if present. If there are nested parentheses work with the innermost pair first.

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3. Complete the column under the connective within the parentheses, if present. You will use the truth values of the connective in determining the final answer in step 5.

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4. Complete the column under any remaining statements and their negation.

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5. Complete the column under any remaining connectives. The answer will appear under the column determined in step 1.For a conjunction, disjunction, conditional or biconditional, obtain the value using the last column completed on the left side and on the right side of the connective.

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5. (continued)

For a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Circle or highlight the answer column and number the columns in the order they were completed.

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Example 3: Truth Table with a Negation

Construct a truth table for ~(~q ⋁ p).

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Example 3: Truth Table with a NegationConstruct a truth table for ~(~q ⋁ p).

Solutionp q ~ (~q ⋁ p)

TTFF

TFTF

FFTF

FTFT

TTFT

TTFF

False only when p is false and q is true.

2314

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Try This

Construct a truth table for the following:

~p ^q

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Example 7: Use the Alternative Method to Construct a Truth TableConstruct a truth table for ~p ⋀ ~q.

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Example 7: Use the Alternative Method to Construct a Truth TableConstruct a truth table with four cases.Solution

p q

TTFF

TFTF

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Example 7: Use the Alternative Method to Construct a Truth TableAdd a column for ~p ~⋀ q.Use columns ~p and ~q to find ~p ~⋀ q.

Solution

p q

TTFF

TFTF

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~p ~q

FFTT

FTFT

~p ~⋀ q

FFFT

It is true only when ~p and~q are true.


Example 9: Determine the Truth Value of a Compound StatementDetermine the truth value for each simple statement. Then, using these truth values, determine the truth value of the compound statement.

15 is less than or equal to 9.

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Example 9: Determine the Truth Value of a Compound StatementLet

p: 15 is less than 9.q: 15 is equal to 9.

Express “15 is less than or equal to 9” as p ⋁ q.

Both p and q are false.p ⋁ qF F⋁ F

Solution

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Example 9: Determine the Truth Value of a Compound StatementDetermine the truth value for each simple statement. Then, using these truth values, determine the truth value of the compound statement.George Washington was the first U.S. president or Abraham Lincoln was the second U.S. president, but there has not been a U.S. president born in Antarctica.

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Example 9: Determine the Truth Value of a Compound StatementLetp: George Washington was the

first U.S. president.q: Abraham Lincoln was the second

U.S. president.r:There has been a U.S. president

who was born in Antarctica.The statement can be written in

symbolic form as (p ⋁ q) ~⋀ r.

Solution

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Example 9: Determine the Truth Value of a Compound Statementp: George Washington was the first

U.S. president.q: Abraham Lincoln was the second

U.S. president.r:There has been a U.S. president

who was born in Antarctica.The statement is (p ⋁ q) ~⋀ r.p is true, q is false, r is false.Since r is false, ~r is true.

Solution

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Example 9: Determine the Truth Value of a Compound StatementThe statement is (p ⋁ q) ~⋀ r.p is true, q is false, ~r is true.

(p ⋁ q) ~⋀ r(T ⋁ F) T⋀ T T⋀

TThe original compound statement is true.

Solution

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Try This: P. 117 # 58

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Homework

P. 115 # 6-60 (x3)

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.2 Truth Tables for Negation,...

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