Geometry Honors

DEDUCTIVE REASONING

Vocabulary

Inductive Reasoning – reasoning based on observed patterns.

Deductive Reasoning – reasoning based on given statements.

Example of Deductive Reasoning

An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic finds that the battery is dead. What conclusion can he make?

What if he discovers the car won’t start. Can he conclude that the car has a dead battery?

Deductive Reasoning

Law of Detachment– if a conditional statement is true, and its hypothesis is true, then it’s conclusion must be true.Example Utilizing the

Law of DetachmentIf a student wants to go to college, then he/she must study hard. Joe wants to go to UCLA. What conclusion can you draw?

Your turn

The Law of Detachment

A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?

If you make a field goal in basketball, you score two points. Jenna scored two points. Therefore, you conclude Jenna made a field goal.

Can you use the Law of Detachment to

make the following conclusion?

Deductive Reasoning

Law of Syllogism– if pq and qr are both true, then pr is also true.

Example Utilizing the Law of Syllogism

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. What true conclusion can you make?

Your turn

The Law of Syllogism

If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. What conclusion can you draw?

It is possible to use both Laws of Deductive Reasoning in the same situation.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 4,132 miles long. What conclusion can you draw?

Law of Syllogism:

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

Law of Detachment:

and

If a river is more than 4000 miles long, then it is the longest river in the world. The Nile is 4,132 miles long.

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

The Chain Rule and Symbols:

The Chain Rule is used to combine two conditionals of the form pq and qr into pr.

pq: If Jane leaves home late, then she will miss her train.qr: If Jane misses her train, then she will be late for work.

CONCLUSIONpr: If Jane leaves home late, then she will be late for work.

Chain Rule Example

pq: If you have a job, then you will get money.qr: If you have money, then you can buy food.

CONCLUSIONpr: If you have a job, the you can buy food.

Chain Rule Example

More Complex Chain Rule Examples

~ad c ~b a b

What conclusion can we draw?

We can start by writing the contrapositive of the first conditional (because they have the same truth value).

~da

~daNow use the chain rule…

~da a b

Now we need the contrapositive of c ~b…

b ~c CONCLUSION: ~d ~c

Let’s do the Chain Rule Worksheet together.

1. If you are involved in extra-curricular activities, your school spirit increases. If your school spirit increases, then your grades improve.

Draw a conclusion for the following:

2.p qq tr p

Draw a conclusion for the following:

3. ~x y z ~x y ~t

Draw a conclusion for the following:

4.~x z t ~z ~t r

Draw a conclusion for the following:

5. a b~c ~b~a e

Draw a conclusion for the following:

6. e b a ~c~e c~a f

Draw a conclusion for the following:

7. m n t ~n k ~p~m p ~t ~r

Draw a conclusion for the following:

HOMEWORK

Pg. 84: 1-15Logic WorksheetChain Rule WS #2Quiz 2.1-2.3 Mon.