“Indubitably.”
“The proof is in the pudding.”
Le pompt de pompt le solve de crime!"
Deductive Reasoning
Je solve le crime. Pompt de pompt pompt."
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
Effect
Justification
Conclusion
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.
hypothesis
Science: A body in motion will stay in motion
unless acted on by an external force.
The whole statement is the 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.
If a number is divisible by 6, then it is divisible by 2.
Which is the hypothesis?
Which is the conclusion?
Declarative Sentence
No 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.
Note that the subject is the cause and everything else is the effect.
If it is a redwood tree,
then it will grow very tall, very thick and have a long life span.
Pete Rose is banned from playing baseball.
If he is Pete Rose, then he is banned from playing baseball.
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.
When it is raining, people outside get wet.
All dogs go to heaven.
If an animal is a dog, then it will go to heaven.
Only if
Pitfall
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
If x 1, then x 1.
2
x 1 onlyif x 1.
2
Converse
Every 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.
Converse:
If a number is divisible by 2, then it is even.
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.
True
If you live in Pennsylvania, then you live in Philadelphia.
False You could live in Pittsburgh.
If you are citizen of the US, then you were born in the US.
False You could be a naturalized citizen.
If you were born in the US, then you are citizen of the US.
True
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.
Biconditionals
The phrase “if and only if” is used
when the conditional and converse are both
true. They are then called biconditionals.
M is the midpoint of
AB
if and only if
AM = MB and M is between A and B.
The symbol or abbreviation of “if and only if” is
M is the midpoint of
AB


AM = MB and M is between A and B.
More examples of biconditionals.
You are tall if and only if your height is 6 ft. or more.
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.
x 1 if and only if x 1
2
If x 1 , then x 1.
2
If x 1 , then x 1.
2
True
False
Try x = -5
More examples of biconditionals.
2
x is odd if and only if x is odd.
x is odd  x is odd.
2
True
x is even  x is even.
True
x is odd  x is odd.
True
2
3
Counterexamples
Conditional 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.
George Bush lives in Texas,
but he lives on a ranch rather than a city.
Also
Someone could live in San Antonio or Austin.
Find Counterexample
If 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.
20 > 10 but 20 not > 100.
Let x = 20
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 a midpt of AC .
B
A
C
Note 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.
Summary
Conditionals are cause and effect statements.
Conditionals are usually written in an If-then format.
If A, then B.
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.
Or
Make the subject the hypothesis and the rest the conclusion.
This is pitfall to avoid.
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.
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.
M is the midpoint of
AB
if and only if
AM = MB and M is between A and B.
The symbol or abbreviation of “if and only if” is
M is the midpoint of
AB


AM = MB and M is between A and B.
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.