Warm Up

Describe the picture using the geometry vocabulary from chapter 1.Describe the picture using the geometry vocabulary from chapter 1.

Conditionals

Students will find and identify the converse, inverse and contrapositive of a conditional

A conditional statement is an if-then statement.

If this is true, then that is true.

For example: If you fail all of your geometry tests,

then you fail geometry. If a and b are positive integers, then

a+b is a positive integer If line m and line n intersect, then

they have a point in common.

A conditional statement has two parts:The hypothesis (the “if” part)The conclusion (the “then” part)

Example: If you want to make an omelette, then you have to break a few eggs.

Hypothesis: You want to make an omelette.

Conclusion: You have to break a few eggs.

Notation:

Sometimes we write a conditional like this:

p → q

Here p represents the hypothesis and q represents the conclusion.

For example:

p = “It is raining”q = “I need an umbrella”

p → q = “If it is raining, then I need an umbrella.”

When is a conditional true?If every time the hypothesis is true, the conclusion is also true.

When is a conditional false?If it is possible for the hypothesis to be true and the conclusion false.

For example: “If a is an integer, then a is a real number.”

Every integer is a real number, so this conditional is true.

For example: “If b is an integer, then b is a positive number.”

It is possible for b to be an integer and not be a postive number. So this conditional is false.

How do you show that a conditional is false?Find a counterexample.

Example: If you are in Maryland, then you are in Baltimore.Counterexample: All of you are in Maryland, but none of you are in Baltimore.

Example: If a and b are integers, then a+b is greater than either a or b.Counterexample: Suppose a = 5 and b = -1. Then a+b = 4, and 4 < 5.

Important Point: You only need one counterexample to show that a conditional is false.

“Hidden” Conditionals

Conditionals won't always appear in “if-then” form. Sometimes we have to pull them apart and rewrite them.

Example: Every good boy deserves fudge.

If-then form: If someone is a good boy, then he deserves fudge.

Example: You can have dessert if you finish dinner.

If-then form: If you finish dinner, then you can have dessert.

Example: You get an A in Geometry only if you do your homework.

If-then form: If you get an A in geometry, then you do your homework.(Or: If you got an A in geometry, then you did your homework.)

Converse: formed by switching the hypothesis and conclusion.

Example: Statement: If you see lightning, then you hear thunder.

Converse: If you hear thunder, then you see lightning.

Using our notation, if the original conditional is:

p → q

then the converse is:

q → p

A statement can also be altered by a negation by writing the negative of the statement.

Statement Negation

m A=∠ 30 �̊ m A∠ ≠ 30 �̊

Angle A is acute. ____________

The symbol for negation is ~. So “~p” means “not p” or “It is not true that p.”

Inverse: negating hypothesis and conclusion of a conditional statement (negating the conditional)

Example: If it is raining, then I need an umbrella.

Inverse: If it is not raining, then I do not need an umbrella.

Using our notation, if the original conditional is:

p → q

Then the inverse is:

~p → ~q

Contrapositive: negating hypothesis and conclusion of the converse of the conditional (negating the converse)

Example: If it is raining, then I need an umbrella.

Contrapositive: If I do not need an umbrella, then it is not raining.

Using our notation, if the original conditional is:

p → q

Then the inverse is:

~q → ~p

ExamplesConditional: If a is an integer, then a is a real number.

Converse:

Inverse:

Contrapositive:

If a is a real number, then a is an integer.

If a is not an integer, then a is not a real number.

If a is not a real number, then a is not an integer.

Question: Which of these are true?

The conditional and contrapositive are true, but the converse and inverse are not.

ExamplesConditional: If I am an American citizen, I was born in Chicago.

Converse:

Inverse:

Contrapositive:

If I was born in Chicago, I am an American citizen.

If I am not an American citizen, I was not born in Chicago.

If was not born in Chicago, I am not an American citizen.

Question: Which of these are true?

The converse and inverse are true, but the conditional and contrapositive are not.

For the following conditionals, write the hypothesis, conclusion, converse, inverse, and contrapositive.

1. If it rains, then the game will be canceled.

2. You will be tired if you don't sleep.

3. x=-2 only if x2=4.

4. If a is divisible by 2, then a is divisible by 4.

5. If x=5, the x+20=25.

6. If b<0, then b2<0.

Now go back and mark each conditional, converse, inverse, and contrapositive as either true or false. What patterns do you see?

Your Turn

Take out a piece of paper.

Make up your own conditional.

Identify the hypothesis and conclusion.

Find the inverse, converse, and contrapositive.

Indicate which (conditional, inverse, converse, and contrapositive) are true, and which are false.