SMP SEGEO C02L02 066-071missbarker316.weebly.com/uploads/3/9/7/5/39757470/smp... · 2020. 8. 12. · 2-x 1. These statements do not contain if or then, but they can easily be rewritten

66 The Language and Logic of Geometry

Lesson

Conditional Statements

Chapter 2

2-2

BIG IDEA Conditional statements, those which can be put in the form “If …, then …,” are the basis of logical thinking in mathematics.

Conditional Statements

A statement is a sentence that is either true or false and not both. “My dog has fl eas” is a statement because either the dog has fl eas or it doesn’t. A compound statement is a sentence that combines two or more statements with some type of connective such as and, or, or if … then. “My dog has fl eas and fl eas make dogs itch” is an example of a compound statement. If a compound statement can be written in if … then form, then the statement is called a conditional statement. Below are several examples of conditional statements.

If the light is red, then you must stop.

If x = y, then x2 = y2.

If you live in Springfi eld, then you live in Illinois.

A fi gure is a polygon if it is a triangle.

You can get your driver’s license if you pass the test.

If I move my queen here, then you will take it with your rook.

The two statements that make up a conditional statement have special names. The statement that follows “if” is called the antecedent, while the other statement, which often follows “then” is called the consequent. (Alternate names for antecedent and consequent are hypothesis and conclusion, respectively.)

If the light is red, then you must stop. antecedent consequent

Both the antecedent and consequent alone would be complete sentences if they began with a capital letter and ended with a period. Notice that this statement can be written as follows without losing any meaning.

You must stop when the light is red. consequent antecedent

Vocabulary

statement

compound statement

conditional statement

antecedent

consequent

instance of a conditional

counterexample to a

conditional

How many triangles are shown in the fi gure below?

Mental Math

The queen has not always

been the strongest chess

piece, nor has it always been

a queen. When chess fi rst

spread to Europe, the piece

next to the king was the ferz

and was the second-weakest

piece on the board.

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Conditional Statements 67

Lesson 2-2

Example 1

Write the antecedent and consequent of the following conditional:

If two unique lines intersect, then they intersect at exactly one point.

Solution The antecedent follows the word if. Antecedent:

Two unique lines intersect. The consequent follows the word then. Consequent: They intersect at exactly one point.

QY1

Rewriting Sentences as Conditionals

There are sentences that have the same meaning as a conditional, but do not contain the words if or then. Some that you have seen in this book are:

Given any two points, there is exactly one line containing them.

The slope of the line containing points (x1, y1) and (x2, y2) is y2 - y1 ______ x2 - x1

.

These statements do not contain if or then, but they can easily be rewritten as conditional statements:

If there are two points, then there is exactly one line containing them.

If the points (x1, y1) and (x2, y2) are on a line, then the slope

of that line is y2 - y1 ______ x2 - x1

.

In addition, sometimes statements follow the pattern All A are B, or Every A is a B. These statements can be rewritten as the conditional If something is an A, then it is a B. In geometry, often the “something” is “a fi gure.”

QY2

Abbreviating Conditionals

Sometimes it is easier to abbreviate a conditional by replacing the statements with letters. We will always use lower case letters to do this. Consider the statement, If I move my queen here, you will take it with your rook. You can let q = I move my queen here and r = you will take it with your rook. Now you can rewrite the statement as If q, then r. A still shorter way of writing this is q � r. The symbol “�” means “implies” and takes the place of “if … then.” Thus, the statement “q � r” is read “q implies r.”

QY3

QY1

Write the antecedent and consequent of the statement: You can get

your driver’s license if you

pass the test.

QY2

Rewrite the following statement as a conditional: All segments

are convex sets.

QY3

Let g = you understand

geometry, and let m = you are a budding

mathematician. Rewrite the statement g � m in English.



Chapter 2

Instances of Conditionals

Consider this statement: If you live in Springfi eld, then you live in Illinois. Suppose Oscar lives in Springfi eld, Illinois. Oscar lives in Springfi eld, so the antecedent is true. He also lives in Illinois, so the consequent is true. Because Oscar is an example of someone who makes both the antecedent and the consequent true, we call him an instance of the conditional.

Defi nition of Instance of a Conditional

An instance of a conditional is a specifi c case in which both the antecedent (if part) and the consequent (then part) of the conditional are true.

Truth and Falsity of Conditionals

A conditional statement is true if, for every possible case in which the antecedent is true, the consequent is also true. It is often hard to show that a conditional is true because you must show that the conditional holds for all cases in which the antecedent is true.

Example 2

Oscar says that because he lives in Springfi eld, Illinois, the above

conditional statement, If you live in Springfi eld, then you live in Illinois, is

true. Is Oscar correct? Justify your answer.

Solution No. Oscar is one instance of the conditional. To prove

the conditional true, he would have to show that everyone from a

place called Springfi eld is also from Illinois.

To prove that a conditional statement is false, all you need is one instance in which the antecedent is true, but the consequent is false. This instance is a counterexample to the conditional.

Defi nition of Counterexample to a Conditional

A counterexample to a conditional is a specifi c case for which the antecedent (if part) of the conditional is true and its consequent (then part) is false.

In Example 2, one person from Springfi eld, Massachusetts, (or any of the 33 cities named Springfi eld in the United States alone that are not in Illinois) would serve as a counterexample to the conditional statement.

According to the 2000 U.S.

Census, about 152,000 of the

people who lived in a place

called Springfi eld lived in

Springfi eld, Massachusetts.



Lesson 2-2

That person lives in Springfi eld, so the antecedent is true, but the person does not live in Illinois, so the consequent is false. Notice that to prove a conditional statement is true requires you to show that it always works in all cases, but proving it false requires only one counterexample!

QY4

Venn Diagrams

A helpful way to analyze a conditional statement is to model it using a Venn diagram. Venn diagrams show relationships among sets.

At the right is a Venn diagram that shows the relationship between squirrels and things that cannot drive cars. It shows that the set of squirrels is a subset of the set of things that cannot drive cars because the set of squirrels is completely contained within the set of things that cannot drive cars. From this diagram we can determine that the statement If you are a squirrel, then you cannot drive a car is true. This is because all squirrels are within the inner circle, which also means they are within the outer circle. However, the statement If you cannot drive a car, then you are a squirrel is not true (obviously). Using the Venn diagram, we can see this because if an item is in the outer circle, it is not necessarily in the inner circle.

QY5

Example 3

Use the Venn diagram at the right to create one true conditional statement

and one false conditional statement about fi ctitious “wibbles”

and “squibbles.”

Solution

True Statement: If it is a squibble, then it is a wibble.

False Statement: If it is a wibble, then it is a squibble.

Questions

COVERING THE IDEAS

1. Fill in the Blank An if-then statement is commonly called a ? statement.

2. Fill in the Blanks The clause following if in a conditional statement is called the ? , while the clause following then is called the ? .

QY4

Provide a counterexample to prove that the following conditional statement is not true: If you go outside without

an umbrella, then you will

get wet.

things that cannotdrive cars

squirrels

wibbles

squibbles

QY5

Provide a counterexample to the statement:If you cannot drive a car,

then you are a squirrel.



Chapter 2

3. Let w = today is Wednesday and t = tomorrow is Thursday. a. Rewrite w � t in English. b. What is the antecedent of the conditional? c. What is the consequent?

4. What must you do to prove that a conditional statement is true?

5. What is a counterexample?

6. Rewrite this statement as a conditional: When you add two even numbers, the sum is another even number.

7. Use the Venn diagram at the right to write a true conditional statement.

APPLYING THE MATHEMATICS

8. In Lesson 1-3, there is the following statement: A network is traversable when it has two or zero odd nodes. Write this as a conditional statement.

9. Rewrite the Unique Line Assumption Postulate from Lesson 1-5 as a conditional statement.

10. Consider the following statement: If x2 = 16, then x = 4. Provide a counterexample to this conditional.

11. Consider this conditional: If today is the 30th day of the month, then tomorrow is the 1st day of the next month.

a. One instance of this conditional is June 30. Give all other instances of this conditional.

b. Provide a counterexample that shows why this conditional is not true.

12. A prime is any number whose only factors are 1 and itself. Suppose your friend says that if n is an even integer, then n2 + 1 is a prime. Find a counterexample that shows that your friend is wrong.

13. Create three true conditional statements about the fi ctitious animals shown in the Venn diagram at the right.

14. Programming languages for computers and calculators often use conditional statements in determining the actions taken by the program. Consider the program at the right.

a. In the program, what happens if you answer the question with a “Yes”?

b. What happens if you answer the question in any other manner?

Ontario

Guelph

denifumps

marklemumps

filimumps



Lesson 2-2

REVIEW

15. Characterize the shaded set of points as convex or nonconvex. (Lesson 2-1)

a. b. c.

16. Is a ray a convex fi gure? (Lesson 2-1)

17. Do you think a ray should be considered a polygon? Why or why not? (Lesson 2-1)

18. In how many points do the lines with equations 2y + 3x = 4 and y = –1.5x + 7 intersect? (Lesson 1-5)

19. Describe the set {–1, 0, 1, 2, 3, . . .} in set-builder notation. (Previous Course)

EXPLORATION

20. Consider the following statement: If n is a positive integer,n2

+ n + 41 is a prime number. a. Choose two different values for n to see if you get a

prime number. b. Does your answer for part a prove that the conditional

statement is true? c. Explain why n = 41 is a counterexample. d. Find another counterexample.

21. A Rube Goldberg machine is a machine or contraption that performs a very simple task in a very complex way. Below is a diagram of one such machine.

DE F

I

HG

C

B

J K

L

A

M

a. Write a conditional statement that summarizes the actions of the machine.

b. What are the antecedent and the consequent of the conditional statement that you wrote?

QY ANSWERS

1. The antecedent is you

pass the test. The consequent is you can get your driver’s

license.

2. If a fi gure is a segment, then it is a convex set.

3. If you understand geometry, then you are a budding mathematician.

4. You go outside when it is not raining and you do not get wet.

5. Answers vary. Sample: A baby is not a squirrel and a baby cannot drive a car.


Documents

SMP SEGEO C02L02 066-071missbarker316.weebly.com/uploads/3/9/7/5/39757470/smp... · 2020. 8. 12. · 2-x 1. These statements do not contain if or then, but they can easily be rewritten