2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
Deductive reasoning is a_______________________________________________________ process of reasoning logically from given facts to a conclusion.
pq p q pq qr p r
The Law of Syllogism allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement.
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
q A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? p
pq p q
“it rains”. The first sentence contains a conditional statement. The hypothesis is ____________ the garden will be watered. Because the hypothesis is true, the gardener can conclude that ________________________
Suppose that a mechanic begins work on a car and finds that the car will not start. Can the mechanic conclude that the car has a dead battery? Explain.
No, there could be other things wrong with the car, such as a faulty starter.
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
For given statements, what can you conclude? Given: If A is acute, then mA 90. A is acute.
p
q
pq p q
p
A conditional and its hypothesis are true. By the Law of Detachment , you can conclude that the conclusion, mA 90 is true.
Vladimir Nuñez should not pitch a complete game on Tuesday.
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
Use the Law of Syllogism to draw a conclusion from the following true statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. The conclusion of the first conditional is the hypothesis of the second conditional. This means you can apply the Law of Syllogism. The Law of Syllogism: If then
p r
p q
is a true statement.
and
q r
are true statements, pq qr p r
So you can conclude: it is a rectangle. If a quadrilateral is a square, then ________________________________
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
If possible, state a conclusion using the Law of Syllogism. If it is not possible to use this law, explain why. a. If a number ends in 0, then it is divisible by 10. If a number is divisible by 10, then it is divisible by 5. If a number ends in 0, then it is divisible by 5.
b. If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2. Not possible, the conclusion of one statement is not the hypothesis of the other statement.
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism.
Drawing Conclusions Use the Laws of Detachment and Syllogism to draw a possible conclusion. If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman. The circus is in town. Because the conclusion of the first statement is the you can apply the
Law of Syllogism
hypothesis
of the second statement,
to write a new conditional:
If the circus is in town, then ____________________________ Paul is working as a night watchman . The third statement means that the hypothesis of the new conditional is true. You can use the
Law of Detachment
Paul is working as a night watchman .
to form the conclusion:
2-3
Deductive Reasoning Objectives: 1) To use Law of Detachment. 2) To use Law of Syllogism. Use the Laws of Detachment and Law of Syllogism to draw a possible conclusion. The Volga River is in Europe. If a river is less than 2,300 miles long, it is not one of the world’s ten longest rivers. If a river is in Europe, then it is less than 2,300 miles long.
Conclusion: If a river is in Europe, it is not one of the world’s ten longest rivers. Conclusion:
The Volga River is not one of the world’s ten longest rivers.
2-3
Deductive Reasoning
page 92 textbook
#47. Reasoning In a band, Amy, Bob, and Carla are the drummer, guitarist, and keyboard player. Use the clues to find the instrument that each one plays. 1. 2. 3.
Carla and the drummer wear different-colored shirts. The keyboard player is older than Bob. Amy, the youngest band member, lives next door to the guitarist.
You can solve this type of logic puzzle by eliminating possibilities. Copy the grid below. Put an X in a box once you eliminate it as a possibility.
Amy plays the drums. Bob plays the guitar.
Carla plays the keyboard.
Using a matrix logic: (another example) Ted, Ken, Allyson, and Jane (two married couples) each have a favorite sport: running, swimming, biking, and golf. Given the following clues, determine who likes which sport: 1. Ted hates golf. He agrees with Mark Twain that golf is nothing but a good walk spoiled. 2. Ken wouldn’t run around the block if he didn’t have to, and neither would his wife. 3. Each woman’s favorite sport is featured in a triathlon. 4. Allyson bought her husband a new bike for his birthday to use in his favorite sport. Use the following matrix to work out this problem. Use an X to represent “no” and an O to represent “yes”. SPORTS Running
ANSWER: Ted is the biker, Ken is the golfer Allyson is the runner, and Janie is the swimmer.
Swimming Biking
Golf
Ted
Ken
X
Allyson
X2
O
X2 O
X O4
X X4
X X4
X1
O
X3
GEOMETRY/C.Bautista
Janie
9/15/2013
X4 X3
2-3
Deductive Reasoning
(More reasoning examples)
Does each conclusion use inductive reasoning or deductive reasoning? 1. The sum of the angle measures of a triangle is 180°. Two angles of a triangle are 40° and 60°, so Kandy concludes that the third angle measures 80°. deductive reasoning 2. All of the students in Raul’s Geometry class are sophomores. Alex takes Geometry, but has another teacher. Raul concludes that Alex is also a sophomore. inductive reasoning 3. A detective learns that his main suspect was out of town the day of the crime. He concludes that the suspect is innocent. deductive reasoning 4. The Mighty Pirates scored over 75 points in each of ten straight games. The newspaper predicts that they will score more than 75 points tonight. inductive reasoning 5. You observe that for the last 5 of 6 weeks, the school cafeteria has served chicken on Thursdays. Since tomorrow is Thursday, you conclude that the cafeteria will serve chicken. inductive reasoning 6. Since your lowest grade in Algebra 2 on any assignment or test is 92, you conclude that your average grade in Algebra 2 class is 92 or above. deductive reasoning
2-3
Deductive Reasoning
page 98 textbook
For Exercises 27–31, use the cartoon and deductive reasoning to answer yes or no. If no, explain. 27. Is a person with a red car allowed to park here on Tuesday at 10:00 A. M.? No, red cars can never park. 28. Is a man with a beard allowed to park here on Monday at 10:30 A. M.? No, guys with beards cannot park on Monday. 29. Is a woman with a wig allowed to park here on Saturday at 10:00 A. M.?
yes
30. Is a person with a blue car allowed to park here on Tuesday at 9:05 A. M.? No, there is no parking Tuesday from 6:49 A.M. to 9:11 A.M.
31. Is a person with a convertible with leather seats allowed to park here on Sunday at 6:00 P.M.?
yes
2-3
Deductive Reasoning
page 98 textbook
32. Reasoning. Assume that the following statements are true. 9. Suppose only one Only two of the four students If Anita goes to the concert, Beth will go. student went to the went to the concert. If Beth goes to the concert, Aisha will go. concert, which student Who were they? If Aisha goes to the concert, Ramon will go. was that? Ramon Read and Plan 1. The Law of Detachment requires a ______________ conditional of a statement with a true _______________. hypothesis 2. Is it appropriate to use the Law of Detachment to solve the problem? ________ YES Plan and Solve 3. In fact it is appropriate to use the Law of Detachment several times. Let “Anita goes to the concert” be Beth will go. the true hypothesis, and apply the Law to the first conditional. What is the true conclusion? _____________ 4. Continue applying the Law using your answer from part 3 as a new true hypothesis. If Anita went to the Anita, Beth, Aisha, and Ramon concert, of the four students who went to the concert? _______________________________ 5. According to the problem statement, is this what happened? _____ NO Why? __________________________ Only two students went. 6. Assume that Anita did not go but that Beth went to the concert. Under these assumptions, which of the four students went to the concert? ______________________ Beth, Aisha, and Ramon Is this what happened?____________________ No, only two went. 7. Which two of the four students went to the concert?__________________ Aisha, and Ramon Look Back and Check 8. Does your answer seem reasonable? For instance, is it possible for another pair of these four students to go Yes, it is reasonable. It is not possible for another pair to go to the concert. to the concert? _________________________________________________________________________
