3.2
Truth Tables for Negation, Conjunction, and Disjunction
Copyright © 2005 Pearson Education, Inc. Truth Table
A truth table is used to determine when a compound statement is true or false.
Copyright © 2005 Pearson Education, Inc. Slide 3-2 Conjunction Truth Table Click on speaker for audio
The symbol ^ is read as “and”
p q pq∧
Case 1 T T T
Case 2 T F F
Case 3 F T F
Case 4 F F F
The conjunction is true only when both p and q are true.
Copyright © 2005 Pearson Education, Inc. Slide 3-3 Disjunction
Click on speaker for audio The symbol V is read as “or”
p q p ∨ q
Case 1 T T T
Case 2 T F T
Case 3 F T T
Case 4 F F F The disjunction is true when either p is true,qis true, or both p and q are true.
Copyright © 2005 Pearson Education, Inc. Slide 3-4 Making a truth table
Let’s construct a truth table for p v ~q. This is read as “p or not q”. Step 1: Make a table with different possibilities for p and q .There are 4 different possibilities.
p q Case 1 T T Case 2 T F Case 3 F T Case 4 F F
Copyright © 2005 Pearson Education, Inc. Slide 3-5 Making a truth table (cont’d)
Click on speaker for audio Step 2: Now, make a column for ~q (“not” q) since we want to ultimately find p v ~q
p q ~q Case 1 T T F Case 2 T F T Case 3 F T F Case 4 F F T
Copyright © 2005 Pearson Education, Inc. Slide 3-6 Making a truth table (cont’d) Step 3: Next, make a column for p v ~q. Use the first and third columns to decide the truth values for p v ~q The truth table is now finished.
p q ~q p v~q
T T F T
T F T T
F T F F
F F T T
Copyright © 2005 Pearson Education, Inc. Slide 3-7 Check it out
Use the truth table above to decide the truth value of p V ~q if p is false and q is true.
(Answer: p V ~q is false)
Copyright © 2005 Pearson Education, Inc. Slide 3-8 Next steps
Read Examples 1-6
Work problems 5-13,odds; 43-50,all; 51-54,all from p. 115 in text
Do online homework for Sec . 3.2
Copyright © 2005 Pearson Education, Inc. Slide 3-9 3.3
Truth Tables for the Conditional and Biconditional
Copyright © 2005 Pearson Education, Inc. Conditional
Click on speaker for audio p -> q is read as “if p then q” p q p → q Case 1 T T T Case 2 T F F Case 3 F T T Case 4 F F T
The conditional statement p → q is true in every case except when p is a true statement and q is a false statement.
Copyright © 2005 Pearson Education, Inc. Slide 3-11 Biconditional
Click on speaker for audio
The biconditional statement, pq ↔ means that pq → and qp → , or, symbolically (pq→∧→) ( qp). Take these 2 columns to get column 7
p q (p → q) ∧ (q → p) case 1 T T T T T T T T T case 2 T F T F F F F T T case 3 F T F T T F T F F case 4 F F F T F T F T F
order of steps 1 3 2 7 4 6 5
Copyright © 2005 Pearson Education, Inc. Slide 3-12 Next steps
Read Examples 1 and 5 only
Work problems 7-13, odds; 71-76, all from p. 125 in text
Do online homework for Sec . 3.3
Do online quiz for 3.2 and 3.3
Copyright © 2005 Pearson Education, Inc. Slide 3-13