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Section 2-4: Deductive Reasoning

- Objectives:
- Use the Law of Detachment
- Use the Law of Syllogism

COMPARE AND CONTRAST

Inductive Reasoning: based on observing what has happened and make a conjecture on what will happen

Deductive Reasoning: process of reasoning logically from given statements to a conclusion. If the statements are true, then deductive reasoning produces a true conclusion.

Law of Detachment

Law of Syllogism

versus

Law of Detachment

- If a conditional is true and its hypothesis is true, then its conclusion is true.
- In symbolic form: If p q is a true statement and p is true, then q is true.

Given:

If M is the midpoint of a segment, then it divides the segment into two congruent segments.

M is the midpoint of AB.

You are given that a conditional and its hypothesis are true.

(The statements following the word GIVEN are considered true versus statements that are assumed to be true like axioms and postulates)

By the Law of Detachment, you can conclude that M divided AB into two congruent segments . . . . Or . . . . Symbolically AM ≅MB.

Do Now!

Law of Detachment

For the given statements, what can you conclude?

Given: If A is acute, mA < 90. A is acute.

A conditional and its hypothesis are both given as true.

By the Law of Detachment, you can conclude that the conclusion of the conditional mA < 90 is also true.

Do Now!

Law of Detachment

Does the following argument illustrate the Law of Detachment?

Given:

If you make a field goal in basketball, you score two points.

Jenna scored two points in basketball.

You conclude: Jenna made a field goal.

The two given statements mean that a conditional and its conclusion are both true.

The Law of Detachment applies only if a conditional and its hypothesis are true.

You can make no conclusion. You cannot conclude that Jenna made a field goal.

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.
- Must have TWO If-then statements
- In symbolic form:
- If p q and q r are true statements, then p r is true.

Two True Statements:

If a number is prime, then it does not have repeated factors.

If a number does not have repeated factors, then it is not a perfect square.

The conclusion of one is the hypothesis of the other . . . .

Soooooooo

By the Law of Syllogism, you can conclude that

If a number is prime, then it is not a perfect square.

Try it!

If possible, state a conclusion using the Law of Syllogism. If it is not possible to use this law explain why.

Two True Statements:

If a number ends in zero, then it is divisible by 10.

If a number is divisible by 10, then it is divisible by 5.

Is the conclusion of one the hypothesis of the other?

Yes

Write the conclusion:

If a number ends in zero, then it is divisible by 5.

Try it!

If possible, state a conclusion using the Law of Syllogism. If it is not possible to use this law explain why.

Two True Statements:

If a number ends in six, then it is divisible by 2.

If a number ends in four, then it is divisible by 2.

Is the conclusion of one the hypothesis of the other?

NO

Not possible because…

The conclusion of one statement is not the hypothesis of the other.

Real-world Geography:

Use the Law of Detachment and the Law of Syllogism to draw conclusions from the following true statements.

True Statements:

If a river is more than 4000 miles long, then it is longer than the Amazon.

If a river is longer than the Amazon, then it is the longest river in the world.

The Nile is 4132 miles long.

Can you use the Law of Syllogism? If so, what is the conclusion?

Conclusion:

If a river is more than 4000 miles long, then it is the longest river in the world.

Can you use the Law of Detachment? If so, what can you also conclude?

Conclusion:

The Nile is the longest river in the world.

Students

Venn Diagrams and Quantifiers

SOME

All

None

Some students are aliens.

All students are aliens.

If you are a student, then you are not an alien.

Aliens

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