Chapter 3 Section 3 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

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Chapter 3

Logic

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WHAT YOU WILL LEARN• Truth tables for conditional

statements, and biconditional statements

• Self-contradictions, tautologies, and implications

• Equivalent statements, De Morgan’s law, and variations of conditional statements

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Section 3

Truth Tables for theConditional and Biconditional

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Conditional

The conditional statement p q is true in every case except when p is a true statement and q is a false statement.

TFFCase 4

TTFCase 3

FFTCase 2

TTTCase 1

qp p q

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Biconditional

The biconditional statement, p↔q means that p q and q p or, symbolically (p q) (q p).

5647231order of steps

FTFTFTFFFcase 4

FFTFTTFTFcase 3

TTFFFFTFTcase 2

TTTTTTTTTcase 1

p)(qq)(pqp

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Example: Truth Table with a Conditional

Construct a truth table for ~p ~q.

Solution: Construct standard four case truth table.

p q ~p ~q

TTFF

TFTF

FFTT

TTFT

FTFT

Then fill-in the table in order, as follows:

It’s a conditional, the answer lies under the .231

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Self-Contradiction

A self-contradiction is a compound statement that is always false.

When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction.

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Tautology

A tautology is a compound statement that is always true.

When every truth value in the answer column of the truth table is true, the statement is a tautology.

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Implication

An implication is a conditional statement that is a tautology.

The consequent will be true whenever the antecedent is true.

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Section 4

Equivalent Statements

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Equivalent Statements

Two statements are equivalent if both statements have exactly the same truth values in the answer columns of the truth tables.

In a truth table, if the answer columns are identical, the statements are equivalent.

If the answer columns are not identical, the statements are not equivalent.

Sometimes the words logically equivalent are used in place of the word equivalent.

Symbols: or

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De Morgan’s Laws

~ (p q) ~ p ~ q

~ (p q) ~ p ~ q

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Example: Using De Morgan’s Laws to Write an Equivalent Statement

Use De Morgan’s laws to write a statement logically equivalent to “Benjamin Franklin was not a U.S. president, but he signed the Declaration of Independence.”

Solution: Let

p: Benjamin Franklin was a U.S. president

The statement symbolically is ~p q.

q: Benjamin Franklin signed the Declaration of Independence

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Example: Using De Morgan’s Laws to Write an Equivalent Statement (continued)

Therefore, the logically equivalent statement to the given statement is:“It is false that Benjamin Franklin was a U.S. president or Benjamin Franklin did not sign the Declaration of Independence.”

The logically equivalent statement in symbolic form is

~ p q ~ p ~ q

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To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same.

To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same.

Switching Between a Conditional and a Disjunction

p q ~p q

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Variations of the Conditional Statement

The variations of conditional statements are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

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“if not q, then not p”~p~qContrapositive of

the conditional

“if not p, then not q”~q~pInverse of the conditional

“if q, then p”pqConverse of the conditional

“if p, then q”qpConditional

ReadSymbolic FormName

Variations of the Conditional Statement