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Chapter 3 Logic

WHAT YOU WILL LEARN • Truth tables for negations, conjunctions, and disjunctions

Truth Tables for Negation, Conjunction, and Disjunction Section 2 Truth Tables for Negation, Conjunction, and Disjunction

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 F T

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

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

General Procedure for Constructing Truth Tables 1. Determine if the statement is a negation, conjunction, and disjunction The answer to the truth table appears under: ~ if it is a negation  if it is a conjunction V if it is a disjunction

General Procedure for Constructing Truth Tables (continued) 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 first. 3. Complete the column under the connective within parentheses, if present. You will use the truth values of the connective in determining the final answer in step 5. 4. Complete the column under any remaining statements and their negation.

General Procedure for Constructing Truth Tables (continued) 5. Complete the column under any remaining connectives. 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. For a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Number the columns in the order they were completed.

Example: Construct a Truth Table Construct a truth table for p  ~q. Solution: Construct a standard four case truth table. Then fill-in the table in order, as follows: p q  ~ T F T F T F F T F T T F 1 4 3 2 True only when p is true and q is false.

Example: Truth Table with a Negation and Conjunction Construct a truth table for ~(~q  p). Solution: Construct a standard four case truth table. Then fill-in the table in order, as follows: p q ~ (~q  p) T F T F T F F T F T T F 4 1 3 2 False only when p is true and q is false.