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. a.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. b.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. a.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. a.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 = 10The sum, 10, is even. b.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. a.If you are using the Internet, then you own a computer. You could use the Internet on a computer at a library. b.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 = 2b. x = 0.25, y = 0.25 c. x = 1.2, y = 0.6d. 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.