Unit 2.2 Notes

Unit 2.2 NotesDeductive Reasoning

Vocabulary

1. Deductive reasoning2. Inductive reasoning3. Law of Detachment4. Law of Syllogism

Another way to read pq is "p implies q".

What does pq mean?

It means if “hypothesis”, then “conclusion”.

Inductive Reasoning

Reasoning that is based on patterns

you observe.

2, 4, 6, 8, …

Deductive Reasoning

The process of reasoning logically

from givenstatements to a

conclusion.

Example 1An auto mechanic knows that if a

car has a dead battery, then it will not start.

Deductive Reasoning

The mechanic is working on a Camaro and discovers that it has a dead battery.

What can the mechanic conclude?

Example 1An auto mechanic knows that if a

car has a dead battery, then it will not start.

Deductive Reasoning

The mechanic is working on a Camaro and discovers that it has a dead battery.

Conclude: The Camaro will not start.

Example 1 - ConverseDeductive Reasoning

The mechanic is working on a Camaro and discovers that it will not start.

Can the mechanic conclude the Camaro’s battery is dead?

No, it could be any number of other problems.

Law of Detachment

If pq is a true statement and p is

true, then q must be true.

Example 1Given: lf M is the midpoint of a segment,

then it divides the segment into two congruent segments.

M is the midpoint of AB.

Law of Detachment

What can we conclude?

Example 1Given: lf M is the midpoint of a segment,

then it divides the segment into two congruent segments.

M is the midpoint of AB.

Law of Detachment

Conclude: M divides the segment into two congruent segments, AM and MB

Example 2Given: If a baseball player is a pitcher, then he should not pitch a complete game two days in a row.

Vladimir Nunez is a pitcher. On Monday, he pitches a complete game.

Law of Detachment

What can we conclude?

Example 2Given: If a baseball player is a pitcher, then he should not pitch a complete game two days in a row.

Vladimir Nunez is a pitcher. On Monday, he pitches a complete game.

Law of Detachment

Vladimir should not pitch a complete game on Tuesday.

Does the following argument illustrate the Law of Detachment?

Given: If it is snowing, then the temperature is less than or equal to 32F.

The temperature is 20F.

You conclude: It must be snowing.

Law of Detachment

Does the following argument illustrate the Law of Detachment?

Given: If it is snowing, then the temperature is less than or equal to 32F.

The temperature is 20F.

You conclude: It must be snowing.

Law of Detachment

No. This statement is not true. Truth Value: False

If possible use the Law of Detachment to draw a conclusion.

Given: If a road is icy, then the driving conditions are hazardous.

Driving conditions are hazardous.

Law of Detachment

Conclusion?

If possible use the Law of Detachment to draw a conclusion.

Given: If a road is icy, then the driving conditions are hazardous.

Driving conditions are hazardous.

Law of Detachment

Conclusion?

No conclusion. There are other reasons for hazardous driving

conditions.

Law of Syllogism

lf pq and qr are true statements, then pr is

a true statement.

Example 1lf a number is prime, then it does not have repeated factors.

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

Law of Syllogism

What can we conclude?

Example 1lf a number is prime, then it does not have repeated factors.

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

Law of Syllogism

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

Example 2lf a number ends in 0, then it is divisible by 10.

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

Law of Syllogism

What can we conclude?

Example 2lf a number ends in 0, then it is divisible by 10.

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

Law of Syllogism

Conclude: If a number ends in 0, then it is divisible by 5.

Example 3lf a number ends in 6, then it is divisible by 2.

lf a number ends in 4, then it is divisible by 2.

Law of Syllogism

What can we conclude?

Cannot conclude anything.

Example 3-Alf a number ends in 6, then it is divisible by 2.

lf a number is divisible by two, then it is even.

Law of Syllogism

What can we conclude?

Conclude: If a number ends in 6, then it is even.

Example 4lf a river is more than 4000 mi long, then it is longer than the Amazon.

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

The Nile is 4132 miles long.

Law of Syllogism

What can we conclude?

Example 4lf a river is more than 4000 mi long, then it is longer than the Amazon.

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

The Nile is 4132 miles long.

Law of Syllogism

Conclude: The Nile is the longest river in the world.

Properties of EqualityAddition Property If a = b, then a + c = b + c

Subtraction Property If a = b, then a – c = b – c

Multiplication Property If a = b, then a c = b c

Division Property If a = b and c 0, then a/c = b/c

Reflexive Property a = a

Symmetric Property If a = b then b = a

Transitive Property If a = b and b = c, then a = c

Substitution Property If a = b, then b can replace a in an expression.

Distributive Property a(b+c) = ab + ac

Properties of Congruence

Reflexive Property AB AB

Symmetric Property If AB CD, then CD AB

Transitive Property If AB CD and CD EF, then AB EF