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