# 2.4 Notes: Deductive Reasoning

Geometry (H) Lesson 2.4

**2.4 Notes: Deductive Reasoning**

**Lesson Objectives:** In this lesson we will learn how to logically draw conclusions based on certain facts. You should be able to do the following:

ü Use the Law of Detachment to draw a conclusion, if possible

ü Use the Law of Transitivity to draw a conclusion, if possible

ü Use both the Law of Detachment and the Law of Transitivity to draw a conclusion, if possible

**Deductive Reasoning**

Deductive reasoning (sometimes called logical reasoning) is the process of reasoning logically from given statements or facts to a true conclusion. We will be learning two laws of reasoning that will allow us to draw logical conclusions from a set of given facts.

**Law of Detachment**

Example 1: Suppose the following is true...

If you go to Pike Place Market, then you go downtown.

Today you’re going to Pike Place Market.

What conclusion can you draw?

Conclusion:______

The above example illustrates what is known as the Law of Detachment.

**Law of Detachment**

To use the Law of Detachment, identify the hypothesis of the given true conditional. If the second given statement matches the hypothesis of the conditional, then you can make a valid conclusion.

1. What conclusion, if any, can you draw based on the following facts?

If I go to Pike Place Market, then I will buy some Rainier cherries.

I am going to Pike Place Market.

Conclusion:______

Example 2: What conclusion, if any, can you draw based on the following facts?

If I go to Pike Place Market, then I will buy some Rainier cherries.

Today, I bought some Rainier cherries.

Conclusion:______

2. What conclusion, if any, can you draw based on the following facts?

If the Giants are leading by 3 runs or less in the 9th inning, Brian Wilson will pitch.

The Giants are leading by 1 run in the 9th.

Conclusion:______

3. What conclusion, if any, can you draw based on the following facts?

If the Giants are leading by 3 runs or less in the 9th inning, Brian Wilson will pitch.

Brian Wilson is pitching.

Conclusion:______

Venn Diagrams

**Law of Transitivity**

Example 3: What conclusions, if any, can you draw based on the following statements?

If Mike gets a chocolate ice cream, then his brother Dave will get chocolate ice cream.

If Dave gets a chocolate ice cream, he will get ice cream all over his face.

Conclusion:______

The example above illustrates the Law of Transitivity.

**Law of Transitivity (or Law of Syllogism).**

4. Use the Law of Transitivity to draw a conclusion based on the following statements, if possible.

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.

Conclusion:______

5. Use the Law of Transitivity to draw a conclusion based on the following statements, if possible.

If a figure is a rectangle, then it is a quadrilateral.

If a figure is a trapezoid, then it is a quadrilateral.

Conclusion:______

6. Use the Law of Transitivity to draw a conclusion based on the following statements, if possible.

If three or more points are collinear, then they are coplanar.

If three or more points are on the same line, then they are collinear.

Conclusion:______

Venn Diagram

**Using Both the Law of Detachment and the Law of Transitivity**

You can use Law of Detachment and the Law of Transitivity together to draw conclusions based on the given facts.

Example 4: Use the Law of Detachment and the Law of Transitivity to draw a conclusion, if possible.

The Volga River is in Europe.

If a river is in Europe, then it is less than 2300 miles long.

If a river is less than 2300 miles long, it is not one of the world’s ten longest rivers.

Conclusion:______

7. Use the Law of Detachment and the Law of Transitivity to draw a conclusion, if possible.

If the squirrels are ecstatic, then they will throw a party.

If the circus comes to town, then the squirrels will be ecstatic.

The squirrels are throwing a party.

Conclusion:______

Assignment 2C / Deductive Reasoning (2 – 4)At the end of this assignment, you should be able to do the following:

o Use the Law of Detachment to draw a conclusion, if possible

o Use the Law of Transitivity to draw a conclusion, if possible

o Use both the Law of Detachment and the Law of Transitivity to draw a conclusion, if possible

o Use a Venn diagram to help determine what, if any, conclusions can be drawn

**Part I: Practice**

1. If possible, use the Law of Detachment to draw a conclusion for given statements. If not possible, write not possible.

a. If a student gets an A on a final exam, then the student will pass the course.

Felix gets an A on the final exam.

Conclusion: ______.

b. If three points are on the same line, then they are collinear.

Points X, Y, Z are on line m.

Conclusion: ______.

c. If you attend U.C. Davis, then you are an Aggie.

Glenda is an Aggie.

Conclusion: ______.

d. If it is the weekend, then I will be surfing.

It is not a weekday.

Conclusion: ______.

e. If a positive integer is divisible by 2, then it is even.

x is not odd.

Conclusion:______.

2. If possible, use the Law of Transitivity to draw a conclusion for the statements below. If not possible, write not possible.

a. If you read a good book, then you enjoy yourself.

If you enjoy yourself, then your time is well spent.

Conclusion:______.

b. If you improve your vocabulary, then you will improve your score on a standardized test.

If you read often, then you will improve your vocabulary.

Conclusion: ______.

c. If a whole number ends in 6, then it is divisble by 2.

If a whole number ends in 4, then it is divisble by 2.

Conclusion:______.

3. Use the Law of Detachment and the Law of Transitivity to draw a conclusion for the given statements below. If it is not possible to draw a conclusion, tell why.

a. If a mountain is the tallest in Alaska, then it is the tallest in the United States.

If an Alaskan mountain is over 20,300 feet high, then it is the tallest in Alaska.

Alaska’s Mt. McKinley is 20,320 feet tall.

Conclusion: ______.

b. If you live in Manhattan, then you live in New York City.

If you live in SoHo, then you live in Manhattan.

If you live in New York City, then you live in the United States.

Kenneth lives in Soho.

Conclusion:______.

c. If you are studying botany, then you are studying biology.

If you are studying biology, then you are studying a science.

Sharice is taking science this year.

Conclusion:______.

**Part II: Application and Problem Solving**

4. What, if anything, can you conclude from the statements given below?

a. If the Mariners win, then Eduardo will jump for joy.

If a Mariner’s pitcher gets credit for a win, then the Mariners won.

Eduardo jumped for joy.

Conclusion: ______.

b. If Grandma bakes cookies, then the neighborhood cats will storm the kitchen.

If Jill comes by to visit, then Grandma will bake cookies.

Grandma baked cookies.

Conclusion: ______.

c. If Jimmy cries, Buster will howl.

Lance bought a car.

If Lance sells his skateboard, then Jimmy will cry.

If Lance buys a car, then he will sell his skateboard.

Conclusion: ______.

5. Assume that the following statements are true:

*If Lafonda goes to the concert, Ramon will go.*

*If Beth goes to the concert, Lafonda will go.*

*If Anita goes to the concert, Beth will go.*

a. Stephanie saw Beth at the concert, therefore she concludes that Anita must have been there as well. Is she correct? Briefly explain.

b. Ichiro saw Beth in the second row! Therefore he began his search for his buddy Ramon. Was Ichiro justified in his search? Briefly explain.

c. Terrance was told that only two of the four actually went to the concert. If this is true, which two people actually attended the concert?

6. Assume the following statements are true:

I. If Maria drinks juice, then it is breakfast time.

II. If it is lunchtime, then Kira drinks milk and nothing else.

III. If it is mealtime, then Curtis drinks water and nothing else.

IV. If it is breakfast time, then Julio drinks juice and nothing else.

V. Maria drinks juice.

Use only the information given above. For each statement, write *must be true, may be true, or is not true.*

**a. Julio drinks juice. b. Curtis drinks water. c.** Kira drinks milk.

**d. Curtis drinks juice. e. Maria drinks water. f.** Julio drinks milk.

7. Consider the following given statements and conclusion.

Given: If an animal is a fish, then it has gills.

A turtle does not have gills.

You Conclude: A turtle is not a fish.

a. Make a Venn diagram to illustrate the given information.

b. Use the Venn diagram to help explain why the argument uses good reasoning.