1.7 Logical Reasoning

1. 1.7 Logical Reasoning • Conditional statements – written in the form If A, then B. • Statements in this form are called if-then statements. • Ex. If the popcorn burns, then the heat was too high or the kernels heated unevenly. • The part of the statement immediately following the word if is called the hypothesis. • The part of the statement immediately following the word then is called the conclusion.

2. Identify Hypothesis and Conclusion • Identify the hypothesis and conclusion of each statement. • If it is Friday, then Madison and Miguel are going to the movies. Hypothesis: it is Friday Conclusion: Madison and Miguel are going to the movies. • If 4x + 3 > 27, then x > 6. Hypothesis: 4x + 3 > 27 Conclusion: x > 6

3. Conditionals • Sometimes a conditional statement is written without using the words if and then. • A conditional statement can always be rewritten as an if-then statement. • Ex. When it is not raining, I ride my bike. • Rewritten: If it is not raining, then I ride my bike.

4. Write a Conditional in If-Then Form • Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. • I will go to the ball game with you on Saturday. Hypothesis: It is Saturday Conclusion: I will go to the ball game with you If it is Saturday, then I will go to the ball game with you.

5. Write a Conditional in If-Then Form • Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. b. For a number x such that 6x – 8 = 16, then x = 4. Hypothesis: 6x – 8 = 16 Conclusion: x = 4 If 6x – 8 = 16, then x = 4.

6. Deductive Reasoning and Counterexamples • Deductive Reasoning is the process of using facts, rules, definitions, or properties to reach a valid conclusion. • Suppose you have a true conditional and you know that the hypothesis is true for a given case, deductive reasoning allows you to say that the conclusion is true for that case.

7. Deductive Reasoning • 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. • The two numbers are 7 and 3. 7 and 3 are odd, so the hypothesis is true. Conclusion: The sum of 7 and 3 is even. CHECK: 7 + 3 = 10 The sum, 10, is even. • The sum of two numbers is 14. The conclusion is true. If the numbers are 11 and 3, the hypothesis is true also. However, if the numbers are 8 and 6, the hypothesis is false. There is no way to determine the two numbers. Therefore, there is no valid conclusion.

8. Counterexample • To show that a conditional is false, we can use a counterexample. • A counterexample is a specific case in which a statement is false. • It takes only one counterexample to show that a statement is false.

9. Find Counterexample • Find a counterexample for each conditional statement. • If you are using the Internet, then you own a computer. You could use the Internet on a computer at a library. • If the Commutative Property holds for multiplication, then it holds for division. 2 ÷ 1 ≠ 1 ÷ 2

10. Find a Counterexample • Which numbers are counterexamples for the statement below? If x ÷ y = 1, then x and y are whole numbers. a. x = 2, y = 2 b. x = 0.25, y = 0.25 c. x = 1.2, y = 0.6 d. x = 6, y = 3 The only values that prove the statement false are x = 0.25 and y = 0.25. So, these numbers are counterexamples. The answer is B.