2 4 Deductive Reasoning Learning Targets: I understand deductive reasoning I can use the Law of Detachment I can use a Venn diagram to draw conclusions I can use the Law of Syllogism What other evidence might a detective collect? Why is eliminating suspects useful for detectives? What could be some limitations to using fingerprints as evidence? Sep 2 12:58 PM Deductive Reasoning: reasoning using facts, rules, definitions, or properties to reach logical conclusions from given statements. 1. In Miguel's town, the month of April has had the most rain for the past 5 years. He thinks that April will have the most rain this year. 2. Sandra learned that if it is cloudy at night it will not be as cold in the morning as it would be if there are no clouds at night. Sandra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. 3. All of the signature items on the restaurant's menu shown are noted with a special symbol. Kevin orders a menu item that has this symbol next to it, so he concludes that the menu item that he has ordered is a signature item. 4. None of the students who ride Raul's bus own a car. Ebony rides a bus to school, so Raul concludes that Ebony does not own a car. Jul 29 6:35 PM 1
While one counterexample is enough to disprove a conjecture reached using inductive reasoning, it is not a logically correct (or valid) way of proving a conjecture. To prove a conjecture requires deductive reasoning. Law of Detachment is one form of deductive reasoning Jul 29 6:54 PM Jul 29 7:27 PM 2
Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. 5. Given: If a figure is a square, then it is a parallelogram. The figure is a parallelogram. Conclusion: The figure is a square. 6. Given: If three points are noncollinear, they determine a plane. Points A, B, and C lie in place G. Conclusion: Points A, B, and C are noncollinear. 7. Given: If a student turns in a permission slip, then the student can go on the field trip. Felipe turned in his permission slip. Conclusion: Felipe can go on the field trip. Jul 29 7:31 PM Jul 29 7:42 PM 3
Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning using a Venn diagram. 8. Given: If a triangle is equilateral, then it is an acute triangle. The triangle is equilateral. Conclusion: The triangle is acute. 9. Given: If a figure is a square, then it is a polygon. Figure A is a square. Conclusion: Figure A is a polygon. Jul 29 7:55 PM Law of Syllogism: another valid for of deductive reasoning. This law allows you to draw conclusions from two true conditional statements when the conclusion of one statement is the hypothesis of the other. REMEMBER: if the conclusion of the first statement is not the hypothesis of the second statement, no valid conclusion can be drawn. Jul 29 8:06 PM 4
Jul 29 6:59 PM Determine which statement follows logically from the given statements. 10. (1) If Jamal finishes his homework, he will go out with his friends. (2) If Jamal goes out with his friends, he will go to the movies. A. If Jamal goes out with his friends, then he finishes his homework. B. If Jamal finishes his homework, he will go to the movies. C. If Jamal does not go to the movies, he does not go out with his friends. D. There is no valid conclusion. 11. (1) If you do not get enough sleep, then you will be tired. (2) If you are tired, then you will not do well on the test. F. If you are tired, then you will not get enough sleep. G. If you do not get enough sleep, then you will not do well on the test. H. If you do not do well on the test, then you did not get enough sleep. J. There is no valid conclusion. Jul 29 8:13 PM 5
Example 5: Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no conclusion and explain your reasoning. 12. Given: If it snows more than 5 inches, school will be closed. It snows 7 inches. 13. Given: The midpoint divides a segment into two congruent segments. If two segments are congruent, then their measures are equal. M is the midpoint of AB. Jul 29 8:18 PM 6