Lesson 1.7 Logical Reasoning and Counterexamples

1. Lesson 1.7 Logical Reasoning and Counterexamples Objectives: a) To recognize conditional statements. b) To write converses of conditional statements.

2. I. If – Then Statements AKA – Conditional Statements Part following “If” – Hypothesis Part following “Then” – Conclusion Example 1: Conditional: If today is the 1st day of fall, then the month is September. • Hypothesis: • today is the 1st day of fall • Conclusion: • the month is September

3. Example 2: If y = 3 + 5, then y = 8 Hypothesis: y = 3 + 5 Conclusion: y = 8

4. II. Writing Conditionals Example 3: A rectangle has 4 right angles. If a shape is a rectangle then it has 4 right angles. Example 4: An acute angle has a measure less than 90°. If an angle is acute, then it has a measure less than 90°.

5. Not all conditionals are true! If a conditional is true, then every time the hypothesis is true the conclusion will also be true. To prove a conditional is false, you only need to find one counter example. If it is cloudy outside, then it is raining. (T or F) False: Give me a counter example It was cloudy last week but it wasn’t raining.

6. Example 4 and 5 An animal with 4 legs is a dog. Conditional: If an animal has 4 legs then it is a dog. True or False False: Give me a counter example. A

8. III. Converse Flip-flop the hypothesis and the conclusion Example 6: Conditional: If an angle is obtuse, then its measure is more than 90°. (T or F) Converse: If an angle has a measure greater than 90°, then it an obtuse angle. (T or F) If false give a counter example.

9. Example 7: A square has 4 sides. Conditional: If a shape is a square, then it has 4 sides. (Tor F) Converse: If a shape has 4 sides, then it is a square. (T or F) Give me a counter example. A trapezoid

10. Just because the condition is true doesn’t mean the converse is true. Example 8: Condition: If I am asleep, then I am breathing. (T or F) • Converse: • If I am breathing, then I am asleep. (T or F) • Counter Example

11. Notation Conditional: p → q Reads “If p then q” Converse: q → p Reads “If q then p”

12. Conditional Statements Identify the hypothesis and the conclusion: If two lines are parallel, then the lines are coplanar. In a conditional statement, the clause after if is the hypothesis and the clause afterthen is the conclusion. Hypothesis: Two lines are parallel. Conclusion: The lines are coplanar. Quick Check 2-1

13. Write the statement as a conditional: An acute angle measures less than 90º. The subject of the sentence is “An acute angle.” The hypothesis is “An angle is acute.” The first part of the conditional is “If an angle is acute.” The verb and object of the sentence are “measures less than 90°.” The conclusion is “It measures less than 90°.” The second part of the conditional is “then it measures less than 90°.” The conditional statement is “If an angle is acute, then it measures less than 90°.”

14. A counterexample is a case in which the hypothesis is true and the conclusion is false. This counterexample must be an example in which x2 0 (hypothesis true) and x 0 or x < 0 (conclusion false). > > Find a counterexample to show that this conditional is false: If x2 > 0, then x > 0. Because any negative number has a positive square, one possible counterexample is x = –1. Because (–1)2 = 1, which is greater than 0, the hypothesis is true. Because –1 < 0, the conclusion is false. The counterexample shows that the conditional is false.

15. Conditional Statements GEOMETRY LESSON 2-1 Use the Venn diagram below. What does it mean to be inside the large circle but outside the small circle? The large circle contains everyone who lives in Illinois. The small circle contains everyone who lives in Chicago. To be inside the large circle but outside the small circle means that you live in Illinois but outside Chicago.

16. Conditional Converse Hypothesis Conclusion Hypothesis Conclusion x = 9 x + 3 = 12 x + 3 = 12 x = 9 Conditional Statements Write the converse of the conditional: If x = 9, then x + 3 = 12. The converse of a conditional exchanges the hypothesis and the conclusion. So the converse is: If x + 3 = 12, then x = 9.

17. = / The conditional is false. A counterexample is a = –5: (–5)2 = 25, and –5 5. Write the converse of the conditional, and determine the truth value of each: If a2 = 25, a = 5. Conditional: If a2 = 25, then a = 5. The converse exchanges the hypothesis and conclusion. Converse: If a = 5, then a2 = 25. Because 52 = 25, the converse is true.

18. The Mad Hatter states: “You might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!” Provide a counterexample to show that one of the Mad Hatter’s statements is false. The statement “I eat what I see” written as a conditional statement is “If I see it, then I eat it.” This conditional is false because there are many things you see that you do not eat. One possible counterexample is “I see a car on the road, but I do not eat the car.”

19. By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.

20. Writing Math “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” and “p only if q.”

21. Example 2B: Writing a Conditional Statement Write a conditional statement from the following. If an animal is a blue jay, then it is a bird. The inner oval represents the hypothesis, and the outer oval represents the conclusion.

22. Use the following conditional for Exercises 1–3. If a circle’s radius is 2 m, then its diameter is 4 m. 1. Identify the hypothesis and conclusion. 2. Write the converse. 3. Determine the truth value of the conditional and its converse. Show that each conditional is false by finding a counterexample. 4. If lines do not intersect, then they are parallel. 5. All numbers containing the digit 0 are divisible by 10. Hypothesis: A circle’s radius is 2 m. Conclusion: Its diameter is 4 m. If a circle’s diameter is 4 m, then its radius is 2 m. Both are true. skew lines Sample: 105