Sec.2-3Sec.2-3

Deductive ReasoningDeductive Reasoning

Objective:Objective: a) To use the a) To use the Law of Law of Detachment.Detachment.

b) To use the Law b) To use the Law of of Syllogism.Syllogism.

In Ch.1 we used inductive In Ch.1 we used inductive reasoning to predict what will reasoning to predict what will happen next.happen next.

Inductive reasoning is based on Inductive reasoning is based on patterns you observe.patterns you observe.

Today we are going to look at Today we are going to look at the other kind of reasoning.the other kind of reasoning.

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I. Deductive ReasoningI. Deductive Reasoning• AKA….Logical reasoningAKA….Logical reasoning

• System for reaching logical conclusionsSystem for reaching logical conclusions

• Cornerstone of geometry (Proofs)Cornerstone of geometry (Proofs)

• Start w/ a given true statement, then Start w/ a given true statement, then state the next logical statement based state the next logical statement based on the given.on the given.

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Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties. W

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In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning.

Law of Detachment

If p q is a true statement and p is true, then q is

true.

II. Law of DetachmentII. Law of Detachment

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Example 1: Using the Law of Example 1: Using the Law of DetachmentDetachment

• Given: I know if it is raining at my Given: I know if it is raining at my house, then water is being added to my house, then water is being added to my pool.pool.

• Statement: It is raining at my houseStatement: It is raining at my house

• What conclusion:What conclusion:

• Water is being added to my pool.Water is being added to my pool.

Determine if the conjecture is valid by the Law of Detachment.

Example 2B: Verifying Conjectures by Using the Law of Detachment

Given: In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series.

Conjecture: The Red Sox won the 2004 World Series.

Example 2B: Verifying Conjectures by Using the Law of Detachment

Identify the hypothesis and conclusion in the given conditional.

In the World Series, if a team wins four games, then the team wins the series.

The statement “The Red Sox won four games in the 2004 World Series” matches the hypothesis of a true conditional. By the Law of Detachment, the Red Sox won the 2004 World Series. The conjecture is valid.

Another valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other.

Law of Syllogism

If p q and q r are true statements, then p r is a true statement.

III. Law of SyllogismIII. Law of Syllogism

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Determine if the conjecture is valid by the Law of Syllogism.

Example: Verifying Conjectures by Using the Law of Syllogism

Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon.

Conjecture: If a figure is a kite, then it is a polygon.

Example 3A: Verifying Conjectures by Using the Law of Syllogism Continued

Let p, q, and r represent the following.

p: A figure is a kite.

q: A figure is a quadrilateral.

r: A figure is a polygon.

You are given that p q and q r.

Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p r. The conjecture is valid by Law of Syllogism.

Example

Determine if the conjecture is valid by the Law of Syllogism.

Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal.

Conjecture: If an animal is a dog, then it has hair.

Is the conclusion of one statement the conclusion of the other?If so, you can use the law of syllogismY can skip q and go straignt to p

Let x, y, and z represent the following.

x: An animal is a mammal.

y: An animal has hair.

z: An animal is a dog.

Example Continued

You are given that x y and z x.

Since x is the conclusion of the second conditional and the hypothesis of the first conditional, you can conclude that z y. The conjecture is valid by Law of Syllogism.

Example 2: Example 2:

• Given 1: If two planes intersect, then Given 1: If two planes intersect, then they intersect in a line.they intersect in a line.

• Given 2: If two planes are not Given 2: If two planes are not parallel, then they intersect.parallel, then they intersect.

• Statement: Plane ABC and Plane XYZ Statement: Plane ABC and Plane XYZ are not parallel.are not parallel.

• Conclusion:Conclusion:

• The planes intersection is a line.The planes intersection is a line.

What did I learn today?What did I learn today?

• If a figure is a rectangle, then it has If a figure is a rectangle, then it has two pairs of parallel sides.two pairs of parallel sides.

• Figure ABCD is a rectangle.Figure ABCD is a rectangle.• Conclusion: Conclusion: • ABCD has 2 pairs of parallel sides.ABCD has 2 pairs of parallel sides.• Which law did you use to come to Which law did you use to come to

your conclusion?your conclusion?• Law of DetachmentLaw of Detachment

What is your conclusion and What is your conclusion and what law did you use?what law did you use?• If you live in Indianapolis, then you If you live in Indianapolis, then you

live in Indiana.live in Indiana.• Larry lives in Indy.Larry lives in Indy.• If you live in Indiana then you live If you live in Indiana then you live

north of the Ohio River.north of the Ohio River.• Conclusion?Conclusion?• Larry lives north of the Ohio River.Larry lives north of the Ohio River.• Which law(s)Which law(s)• BothBoth