Deductive Reasoning

“The proof is in the pudding.”“Indubitably.”

Je solve le crime. Pompt de pompt pompt."

Le pompt de pompt le solve de crime!"

Conditional Statements

Most of geometric decisions are based on cause and effect statements.

These cause and effect statements are called conditional statements.

These conditional statements are almost always put into an

If-then form.

For example:

If you live in LA, then you live in California.

Cause

ConclusionJustification

Effect

Note: The form is … If A, then B.

A is a cause statement.

B is the effect/conclusion.

The If clause is called the hypothesis.

The then clause is called the conclusion.

Note: In geometry, the hypothesis is the if clause.

This is not to be confused with science.

In science, a hypothesis is an educated guess.

Math: If x is odd, then x2 is odd.

Science: A body in motion will stay in motion unless acted on by an external force.

The whole statement is the hypothesis.

hypothesis

Wordy? Yes

Practice Pick out first the hypothesis, then the conclusion of each statement.

If it is raining, then you will get wet.

If a number is divisible by 2 and divisible by 3, then it is divisible by 2(3).

If x = 5 , then x2 = 25.

If x > 5 , then x +1 > 5.

That was easy! Not so fast.

In English, you can say things in a lot of different manners.

When a number is divisible by 6, it is divisible by 2.

No if and no then.

Which is the cause?

Which is the effect?

Alternate Approach

When a number is divisible by 6, it is divisible by 2.

Reword with if and then.

Which is the hypothesis?

Which is the conclusion?

If a number is divisible by 6, then it is divisible by 2.

Declarative SentenceNo clauses

All Squares have 4 equal sides.

Reword with if and then.

If it is a square, then it has 4 equal sides.

Hypothesis Conclusion

Note that the subject is the cause and everything else is the effect.

Let’s try a few more.

Redwood trees grow very tall, very thick, and have a long life span.

Pete Rose is banned from playing baseball.

If it is a redwood tree,

then it will grow very tall, very thick and have a long life span.

If he is Pete Rose, then he is banned from playing baseball.

Note that the subject is the cause and everything else is the effect.

Conditionals come disguised in various forms.

It is up to you, to be able to find the hypothesis and the conclusion in each one.

Highlight the hypothesis and conclusion of the following.

If the angle is acute, then it is less than 900.

All dogs go to heaven.

When it is raining, people outside get wet.

If an animal is a dog, then it will go to heaven.

Only ifPitfall

When the phrase “only if” is used, the clause containing only if is the conclusion.

If and only if do not mean the same thing.

They are opposite expressions.

Hypothesis Conclusion

21, 1.If x then x

21 1.x only if x

ConverseEvery conditional has a converse.

The converse is made by switching the hypothesis and the conclusion.

Conditional: If a number is even, then it is divisible by 2.

If a number is divisible by 2, then it is even.Converse:

This appears to be easy. It usually is. Let’s try some together.

For each conditional, write the converse.

If you live in Philadelphia, then you live in Pennsylvania.

If you live in Pennsylvania, then you live in Philadelphia.

If you are citizen of the US, then you were born in the US.

If you were born in the US, then you are citizen of the US.

True

False

False

True

You could live in Pittsburgh.

You could be a naturalized citizen.

Note: That if the conditional is true, then converse does not have to be true.Conversely, if the converse is true, then the conditional does not have to be true.

The phrase “if and only if” is used when the conditional and converse are both true. They are then called biconditionals.

The symbol or abbreviation of “if and only if” is

M is the midpoint of if and only if

AM = MB and M is between A and B.

AB

M is the midpoint of

AM = MB and M is between A and B.

AB

Biconditionals

You are tall if and only if your height is 6 ft. or more.

More examples of biconditionals.

21 1if and only ifx x

If you are tall, your height is 6 ft. or more.

If your height is 6 ft. or more , then you are tall.

Notice that it works both ways.

21 , 1.If x then x 2 1 , 1.If x then x

True

FalseTry x = -5

More examples of biconditionals.

2 .if and onx is odd x isy if oddl

2 .x is odd x is odd

2 .x is even x is even3 .x is odd x is odd

True

True

True

CounterexamplesConditional can not be proven true by examples.

Examples only show a conditional true for a finite set of values or conditions. There may be examples that don’t work that you have overlooked.

But you can prove conditionals false by showing an example that doesn’t work.

These examples that don’t work in a conditional are called counterexamples.

Restated counterexamples are example that show a conditional is not always true. Therefore, the statement in unreliable and not true.

Eg The sum of two odd numbers is odd.

3 + 7 = 10

If you live in Texas, then you live in Dallas.

Someone could live in San Antonio or Austin.

George Bush lives in Texas, but he lives on a ranch rather than a city.

Also

Find CounterexampleIf x > 1, then ax >1 Let x = 2 and a = -1

2 > 1 but (-1)(2) = -2 which is not greater than 1.

Notice 1] the conditions of the hypothesis need to be met and 2] the conclusion must be wrong.

If x > 10 , then x > 100. Let x = 20

20 > 10 but 20 not > 100.

If a < 0 and b < 0 , then ab < 0.Let a = -1 and b = -2

-1 < 0 and -2 < 0 but (-1)(-2) = +2 and is not less than 0.

If xy > 0 , then x > 0. Let x = -1 and y = -2

(-1)(-2) = +2 > 0 but x < 0.

, .If C is on AB thenC is on AB��������������

A B C

, .If AB BC then Bis amidpt of ACB

A CNote B is equidistant from A and C but it is not on segment AC.

Remember from the equidistant section, There are an infinite number of points equidistant from 2 points.

Finding a counterexample will not always be easy.

Sometimes you will overlook the situation in which a conditional fails.

With practice and over time, it will become easier and easier.

SummaryConditionals are cause and effect statements.

If A, then B.

Conditionals are usually written in an If-then format.

The if or cause clause is known as the hypothesis.

The then or effect clause is known as the conclusion.

Conditional can be in other forms that need to be converted into if-then format.

When two lines intersect, the intersection is a point.

If two lines intersect, then the intersection is a point.

Squares have four congruent sides.

If a figure is a square, then it has four sides.

OrMake the subject the hypothesis and the rest the conclusion.

When the phrase “only if” is used, the clause containing only if is the conclusion.

If and only if do not mean the same thing.

This is pitfall to avoid.

They are opposite expressions.

If the “cause” and “effect” clauses are reversed, the new statement is called the converse.

Conditional: If A, then B.

Converse: If B, then A.

If one form is true, it does not mean that the other form is necessarily true.

The phrase “if and only if” is used when the conditional and converse are both true.

The symbol or abbreviation of “if and only if” is

M is the midpoint of if and only if

AM = MB and M is between A and B.

AB

M is the midpoint of

AM = MB and M is between A and B.

AB

It is not permissible of logical to prove statements true by example.

You can prove a statement false by showing just 1 example where if does not work. This example is called a counterexample.

C’est fini.

Good day and good luck.