Holt Geometry

2-Ext Introduction to Symbolic Logic2-Ext Introduction to Symbolic Logic

Holt Geometry

Lesson PresentationLesson Presentation

Holt Geometry

2-Ext Introduction to Symbolic Logic

Analyze the truth value of conjunctions and disjunctions.Construct truth tables to determine the truth value of logical statements.

Objectives

Holt Geometry

2-Ext Introduction to Symbolic Logic

compound statementconjunctiondisjunctiontruth table

Vocabulary

Holt Geometry

2-Ext Introduction to Symbolic Logic

Symbolic logic is used by computer programmers, mathematicians, and philosophers to analyze the truth value of statements, independent of their actual meaning.

A compound statement is created by combining two or more statements. Suppose p and q each represent a statement. Two compound statements can be formed by combining p and q: a conjunction and a disjunction.

Holt Geometry

2-Ext Introduction to Symbolic Logic

A conjunction is true only when all of its parts are true. A disjunction is true if any one of its parts is true.

Holt Geometry

2-Ext Introduction to Symbolic Logic

Example 1: Analyzing Truth Values of Conjunctions and Disjunctions

Use p, q, and r to find the truth value of each compound statement.

p: The month after April is May.q: The next prime number after 13 is 17.r: Half of 19 is 9.

A. p q

Both p and q are true, therefore the disjunction is true.

Since r is false the conjunction is false.

B. q r

Holt Geometry

2-Ext Introduction to Symbolic Logic

Check It Out! Example 1

Use p, q, and r to find the truth value of each compound statement.

p: Washington, D.C., is the capital of the United States.

q: The day after Monday is Tuesday.r: California is the largest state in the United States.

A. r pSince p is true the disjunction is true.

B. p q

Since both p and q are true the conjunction is true.

Holt Geometry

2-Ext Introduction to Symbolic Logic

A table that lists all possible combinations of truth values for a statement is called a truth table. A truth table shows you the truth value of a compound statement, based on the possible truth values of its parts.

p q p q p q p q

T T T T T

T F F F T

F T T F T

F F T F F

Holt Geometry

2-Ext Introduction to Symbolic Logic

Make sure you include all possible combinations of truth values for each piece of the compound statement.

Caution

The negation (~) of a statement has the opposite truth value.

Remember!

Holt Geometry

2-Ext Introduction to Symbolic Logic

Example 2: Constructing Truth Tables for Compound Statements

Construct a truth table for the compound statement ~p ~q.

p q ~p ~q ~p ~q

T T F F F

T F F T T

F T T F T

F F T T T

Holt Geometry

2-Ext Introduction to Symbolic Logic

Check It Out! Example 2

Construct a truth table for the compound statement ~u ~v.

u v ~u ~v ~u ~v

T T F F F

T F F T F

F T T F F

F F T T T