1 7 Logical Reasoning and Counterexamples Conditional Statements

1. 7 Logical Reasoning and Counterexamples Conditional Statements (If – then statements) l If there is math class, then there is homework. Hypothesis If there is math class, then there is homework. Conclusion

Example #1: Identify the hypothesis and conclusion of each statement. A) If it is raining, then Beau and Chloe will not play softball. B) If 7 y + 5 < 26, then y < 3.

Check Your Progress with #1 A & B A) If it is warm this afternoon, then we will have the party outside. B) If 8 w – 5 = 11, then w = 2.

Example #2: Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. A) I will go to the ball game with you on Saturday. If it’s Saturday, then I will go to the ball game. B) For a number x such that 6 x – 8 = 16, x = 4 If 6 x – 8 = 16, then x = 4.

Deductive Reasoning: Reaching a valid conclusion based on facts, rules, definitions, or properties Example #3: Determine a valid conclusion that follows from the statement, “if two numbers are odd, then their sum is even” for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why. A) the two numbers are 7 and 3 B) the sum of two numbers is 14 A) 7 & 3 are odd, so the hypothesis is true. There is a valid conclusion. B) The conclusion is true. But if the numbers were 8 & 6, then the hypothesis is false. There is no valid conclusion.

Counterexample: A specific case where the statement is false You only need 1 counterexample to prove a statement false. Provide a counterexample for each conditional statement. A) If Joe did not each lunch, then he must not feel well. Maybe Joe just didn’t have time. B) If the traffic light is red, then the cars must be stopped. Maybe a policeman is there to direct traffic.

Homework Assignment #6 p. 42 16 -42 even, 47 -49