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ConditionalsLessons 2-2, 2-3, 11-1, 11-2

Tuesday, September 3, 13

A conditional statement is a statement written in the form “If..., then...”

“If it is raining, then I will have my umbrella.”

A conditional statement is made up of two parts: the hypothesis and the conclusion.

Symbol: P → Q


Determine the hypothesis and conclusion of each statement.Example 1

If a butterfly has a curved black line on its hind leg , then it is a viceroy.

Hypothesis:Conclusion:

If a number is divisible by four, then it is even.Hypothesis:Conclusion:

A butterfly has a curved black line on its hind legIt is a viceroy

A number is divisible by four

It is even


Rewrite each in if-then form. Underline the hypothesis once and the conclusion twice.

Example 2

A number is an integer if it is a natural number.

The midpoint of a segment bisects the segment.

Two angles that are complementary are acute.

If a number is a natural number, then it is an integer.

If a point is a midpoint, then it bisects the segment.

If two angles are complementary, then they are acute.


Truth ValueThink about this...

“If I get paid, then I will take you to a movie.”

* If you don’t get paid and don’t take me to the movie, have you lied to me?

NO!

A conditional is only false if the hypothesis is true AND the conclusion is false!


True or FalseA conditional is only TRUE if the conclusion is true every time the hypothesis is true.

A conditional is FALSE if there is even one instance when the hypothesis is true and the conclusion is false.

counterexampleTuesday, September 3, 13

Example 3

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

If a student has math sixth hour, then they have Mrs. Terrigno as a teacher.

If today is Wednesday, then tomorrow is Thursday.

Disprove each of the following statements by providing a counterexample.


Using ConditionalsThe Law of Detachment states that, if a conditional P ⇒ Q is true and a case arises where the hypothesis (P) is true, then we can conclude that the conclusion (Q) is also true.

EXAMPLE:If two segments are congruent, then the segments

have the same length.This is a true conditional

Given: AB ≅ XYThis is a case of the hypothesis

Therefore... AB and have the same length.Tuesday, September 3, 13

Be Careful!!In order for the Law of Detachment to work, you must have a true hypothesis, NOT a true conclusion!

If two angles are vertical angles, then they are congruent.

Angle A and Angle B are congruent.

Therefore... Angles A and B are vertical angles

INVALID!!


Using ConditionalsThe Law of Transitivity states that, if a conditional P ⇒ Q is true and a conditional Q ⇒ R is true, then the conditional P ⇒ R is also true.

EXAMPLE:If a team wins 10 games, then they get to play in the finals.

If a team plays in the finals, then they travel to Boston.These are true conditional

The Ravens will travel to Boston.Therefore...

Given: The Ravens won 10 games


Be Careful!!The order of the statements matter when you use the Law of Transitivity!


Example 4If you leave your headlights on overnight, then your car battery will drain. If your car battery trains, then your car might not start. Alex left his headlights on last night.

If you multiply the length and width of a rectangle, you will find the area. The length of a rectangle is 6 and the width of the rectangle is 5.

If you want to be popular, you must dress well. If you dress well, you must wear Brand X jeans.

If a person goes to the circus, then they will see clowns. Sally sees clowns.

What conclusion can be drawn in each of the following situations?

Alex’s car might not start.

The area of the rectangle is 30.

If you want to be popular, you must wear Brand X jeans.

NO CONCLUSIONTuesday, September 3, 13

ConversesThe CONVERSE of a conditional is formed by exchanging the hypothesis and conclusion.

Conditional:Converse:

P ⇒ QQ ⇒ P

Conditional:Converse:

If it is 8:00, then my kids are in bed.

If my kids are in bed, then it is 8:00.

Is a converse always true?Consider the converses of each of the following true statements.

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

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

Converse is TRUE


BiconditionalsSometimes a statement’s converse is true, and sometimes it’s not.

When a conditional and its converse are both true, it can be written as a biconditional statement, using the phrase

‘if and only if.’

Conditional: If a figure is a triangle, then it has three sides.Converse: If a figure has three sides, then it is a triangle

Biconditional: A figure is a triangle if and only if it has three sides.


Example 5

Complete the following statement:

Complete the following statement:

Write the definition above as a biconditional statement.

The midpoint of a segment is a point that divides a segment into two congruent pieces.

If a point is a midpoint, then...

If a point divides a segment into two congruent pieces, then...

A point is a midpoint if and only if it divides a segment into two congruent pieces.

...it is a midpoint.

...it divides a segment into two congruent pieces.


NegationsThe negation of a statement is the opposite of the statement.

Statement: M is the midpoint of AB.

Negation: M is NOT the midpoint of AB.

* The negation of a true statement is false.

* The negation of a false statement is true.

Symbol: ~P


Example 6

Congruent angles have equal measure.

Three points are always coplanar.

A conjecture is not always true.

Write the negation of each statement.

Congruent angles do not have equal measure.

Three points are not always coplanar.

A conjecture is always true.


Related Statements

The CONVERSE is the statement formed by exchanging the hypothesis and the conclusion.

The INVERSE is the statement formed by negating the hypothesis and the conclusion.

The CONTRAPOSITIVE is the statement formed by both exchanging and negating the hypothesis and conclusion.

Remember, a conditional statement is written as “If p, then q.”P → Q

Q → P

~P → ~Q

~Q → ~P


Example 7If a person is 16, then they can get a driver’s license.

Converse:

Inverse:

Contrapositive:

Write the converse, inverse and contrapositive of each statement.

If a person can get a driver’s license, then they are 16.

If a person is not 16, then they cannot get a driver’s license.

If a person cannot get a driver’s license, then they are not 16.Tuesday, September 3, 13

Example 8

If a shape is a triangle, then it has three sides.Converse:

Inverse:

Contrapositive:

Write the converse, inverse and contrapositive of each statement.

If a shape has three sides, then it is a triangle.

If a shape is not a triangle, then it does not have three sides.

If a shape does not have three sides, then it is not a triangle.Tuesday, September 3, 13

Logical EquivalenceIf a conditional is true...

...its converse MIGHT be true.

...its inverse MIGHT be true.

...its contrapositive IS ALWAYS true.

A conditional and its contrapositive are logically equivalent.That means they are either both true or both false.

(By the way, the converse and inverse are also logically equivalent!)


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