Truth Tables for Conditional and Biconditional Statements Section 2.4 Truth Tables for Conditional and Biconditional Statements

Objectives Understand the logic behind the definition of the conditional and the biconditional. Construct truth tables for conditional and biconditional statements. Determine the truth value of a compound statement for a specific case.

Conditional, , if…then Suppose your teacher promises you the following: If you pass the final, then you pass the course. Break the compound statement down into its two component statements. p: You pass the final. q: You pass the course. Two simple statements – 4 possible cases.

Conditional, , if…then p: You pass the final. q: You pass the course. p  q T F NOTE: A conditional is false only when the antecedent is T and the consequent is F.

Example 1: Construct truth table. ~p  q

Example 2: Construct truth table. (p q)  (p q)

Example 3: Construct truth table. r  (p q)

Biconditional, , if and only if Suppose your teacher says the following: You will pass the course, iff you pass the final. Break the compound statement down into its two component statements. p: You will pass the course. q: You pass the final. Two simple statements – 4 possible cases.

Biconditional, , iff p: You will pass the course. q: You pass the course. p q p  q T F NOTE: A biconditional is T only when the component statements have the same value.

Example 4: Construct a truth table. (p  q)  q

Example 5: Construct truth table. (p  ~q)  (q  ~p)

Example 6: Construct a truth table. (p r)  ~(q r)

Key Terms Tautology: a compound statement that is true in all cases these statements are also called “implications”. Self-contradiction: a compound statement that is false in all cases.

Example 7: Determine if the statement is a tautology, self- contradiction, or neither. [(p  q) p]  q

Example 8: Determine if the statement is a tautology, self- contradiction, or neither. [(p  q) ~p]  ~q

Example 9: Determine if the statement is a tautology, self- contradiction, or neither. (p q) →(~p ~q)

Example 10: TB pg. 107/7

Section 2.4 Assignment Classwork: TB pg. 107/2 – 20 Even Must write problem and show ALL work to receive credit for the assignment. NOTE: If your truth table is not complete, then your problem is wrong.