Section 3 2 Truth Tables for Negation Conjunction

What You Will Learn Truth tables for negations, conjunctions, and disjunctions 3. 2 -2

Truth Table A truth table is used to determine when a compound statement is

Negation Truth Table Case 1 Case 2 3. 2 -4 p T F Copyright

Compound Statement Truth Table Case 3. 2 -5 1 2 3 4 p T

Conjunction Truth Table Case 1 2 3 4 p T T F F q

Disjunction Truth Table Case 1 2 3 4 p T T F F q T F p⋁q T T T F The disjunction is true when either p is true, q is true, or both p and q are true. 3. 2 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -8 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Conjunction p ⋀ q is read “p and q. ” p ⋀ q is

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. 3. 2 -10 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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 3. 2 -11 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Truth Table with a Negation Construct a truth table for ~(~q ⋁ p). Solution p q ~ (~q ⋁ p) T T F F F T F 4 F T 1 T T F T 3 T T F F 2 False only when p is false and q is true. 3. 2 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 7: Use the Alternative Method to Construct a Truth Table Construct a truth

Example 7: Use the Alternative Method to Construct a Truth Table Solution Construct a

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 T F F F T T F T F F F T It is true only when ~p and~q are true. 3. 2 -21 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -22 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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 3. 2 -23 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -24 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -25 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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. 3. 2 -26 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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 The original compound statement is true. 3. 2 -27 Copyright 2013, 2010, 2007, Pearson, Education, Inc.