Section 3.2 Truth Tables for Negation, Conjunction, and Disjunction

What You Will Learn Truth tables for negations, conjunctions, and disjunctions

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

Negation Truth Table p ~p Case 1 T F Case 2

Compound Statement Truth Table q Case 1 T Case 2 F Case 3 Case 4

Conjunction Truth Table p q p ⋀ q Case 1 T Case 2 F Case 3 Case 4 The conjunction is true only when both p and q are true.

Disjunction Truth Table The disjunction is true when either p is true, q is true, or both p and q are true. p q p ⋁ q Case 1 T Case 2 F Case 3 Case 4

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.

Conjunction Conjunction p ⋀ q is read “p and q.” p ⋀ q is true only when both p and q are true

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.

Constructing Truth Tables 1. 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

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.

Constructing Truth Tables 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.

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

Constructing Truth Tables 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.

Constructing Truth Tables 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.

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

Example 3: Truth Table with a Negation Construct a truth table for ~(~q ⋁ p). Solution p q ~ (~q ⋁ p) T F T F F T F T T F T F 4 1 3 2 False only when p is false and q is true.

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

Example 7: Use the Alternative Method to Construct a Truth Table Solution Construct a truth table with four cases. p q T F T F

Example 7: Use the Alternative Method to Construct a Truth Table Solution Add a column for ~p ⋀ ~q. Use columns ~p and ~q to find ~p⋀~q. p q ~p ~q ~p ⋀ ~q T F T F F T F T F T It is true only when ~p and~q are true.

Example 9: Determine the Truth Value of a Compound Statement Determine 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.

Example 9: Determine the Truth Value of a Compound Statement Solution Let 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 ⋁ q F ⋁ F F

Example 9: Determine the Truth Value of a Compound Statement Determine 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.

Example 9: Determine the Truth Value of a Compound Statement Solution Let p: 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.

Example 9: Determine the Truth Value of a Compound Statement Solution p: 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.

Example 9: Determine the Truth Value of a Compound Statement Solution The statement is (p ⋁ q) ⋀ ~r. p is true, q is false, ~r is true. (p ⋁ q) ⋀ ~r (T ⋁ F) ⋀ T T ⋀ T T The original compound statement is true.