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Lawyers use logic and reasoning to make convincing arguments in court.

498 UNIT 14 LOGIC AND REASONING

UNIT 14 Logic and Reasoning

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Just as lawyers use logical reasoning to formulate convincing arguments, mathematicians use logical reasoning to formulate and prove theorems. Once you have mastered the uses of inductive and deductive reasoning, you will be able to make and understand arguments in many areas.

Big Idea► Mathematical reasoning involves making convincing arguments to justify or prove

mathematical statements. The result of mathematical reasoning is often a proof. A proof is a convincing argument that conforms to the rules of logic and relies on accepted axioms or already proven results.

Unit Topics ► Reasoning and Argument

► Hypothesis and Conclusion

► Forms of Conditional Statements

► Inductive and Deductive Reasoning

► Analyzing and Writing Proofs

► Counterexample

LOGIC AND REASONING 499



Reasoning and Argument

You use reasoning in daily life and in mathematics.

Suppose a student listening to a news broadcast hears the following. Anchorman to anchorwoman: “Whenever it rains, morning traffic becomes a mess.”Traffic reporter: “Traffic is slow throughout the metropolitan area. All the major roads are jammed.”

The student might reason that it is raining outside. But based on what was actually said, that may or may not be true. It cannot be assumed that because traffic is a mess it is raining. When you understand some basic rules of logic and reasoning, you can avoid making bad assumptions.

Logical reasoning uses statements, which are sentences that tell an idea. A statement can either be true or false.

An argument is a set of statements, called premises, that are used to reach a conclusion.

Syllogism is a logical argument that always contains two premises and a conclusion.

DEFINITIONS

SyllogismsSyllogisms take several forms. The most common syllogism forms are shown in the table, where a, b, and c represent phrases.

Syllogisms

If a, then b.If b, then c.

Therefore, if a, then c.

All a are b.All c are a.∴ all c are b.

If a number is a whole number, then it is an integer.If a number is an integer, then it is a real number.Therefore, if a number is a whole number, then it is a real number.

All integers are real numbers.All whole numbers are integers.Therefore, all whole numbers are real numbers.

The symbol ∴ means therefore.

NOTATION

REASONING AND ARGUMENT 501

(continued)




Example 1 Complete the syllogisms.A. If it is the month of May, then lilacs are blooming. If lilacs are blooming, then tulips are blooming. Therefore, if , then .Solution Identify the parts of the syllogism. Determine that a is “it is the month of May,” b is “lilacs are blooming,” and c is “tulips are blooming.” The completed conclusion is this: Therefore, if it is the month of May , then tulips are blooming . ■

B. All polynomials are mathematical expressions. All are . Therefore, all quadratics are mathematical expressions.Solution Identify the parts of the syllogism. Determine that a is “poly-nomials,” b is “mathematical expressions,” and c is “quadratics.” The com-pleted premise is this: All quadratics are polynomials . ■

Determining if an Argument Is Valid or InvalidAn argument may not be valid. In a valid argument, if the premises are true, then the conclusion must be true. In an invalid argument, it is possible for the conclusion to be false when all the premises are true. Example 2 Determine if the argument is valid or invalid. Explain your answer. If it is a car, then it has an engine.If it has an engine, then you can drive it.Therefore, if it is a car, then you can drive it.Solution The premises are each true: Cars have engines and an engine enables you to drive a car. The conclusion is also true that cars can be driven. However, the argument is flawed. Other factors can make a car immobile. For example, if a car has a flat tire, you cannot drive it even though it has a working engine. Sometimes even though the premise statements in an argument are true, they do not necessarily lead to a true conclusion, as the example above shows. ■

Identifying the Flaw in an ArgumentExample 3 Sean sees four paw prints in the snow and states, “An animal with four legs has walked through here.” Suzy says, “All dogs have four legs, so the animal must be a dog.” What is the flaw in Suzy’s argument?Solution The argument is flawed because Suzy assumes that all animals with four legs are dogs. That statement is not true. Other animals with four legs are not dogs, such as wildebeests, elephants, and cats. ■

The argument has three statements and three terms, but the placement of the terms does not form a valid syllogism.

THINK ABOUT IT



Problem Set

Complete the syllogism.


1. If x is an even number, then x + 1 is an odd number.

If x + 1 is an odd number, then x + 2 is an even number.

Therefore, if , then .

2. If a regular polygon has three sides, then it is a triangle.

If a regular polygon is a triangle, then the sum of the measures of its interior angles is 180°.


3. All students that receive a zero are students that fail.

All students that cheat are students that receive a zero.

Therefore, .

4. If it snows, then school will be canceled.

If school is canceled, then you can sleep in.


5. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

If two lines are parallel, then alternate-interior angles around a transversal that intersects the lines are congruent.


6. If two lines are perpendicular, then the two lines form a right angle.

If the two lines form a right angle, then the two lines meet at a 90° angle.


7. If a measurement is 36 inches, then it is 3 feet.

If , then .

Therefore, if a measurement is 36 inches, then it is 1 yard.

8. All mammals are warm-blooded.

All dogs are mammals.

Therefore, .

9. If it is hot outside, then I will turn on the air conditioning.

If , then .

Therefore, if it is hot outside, the electric bill will be high.

10. If it is November, then the leaves will be falling.

If , then .

Therefore, if it is November, I will have to rake the leaves.




11. If the discriminant of a quadratic function is positive, then the function has two real solutions.

If a function has two real solutions, then it will cross the x-axis twice.


12. If a triangle is isosceles, then the triangle has two congruent sides.

If a triangle has two congruent sides, then the base angles are congruent.


Determine if the argument is valid or invalid.

13. If a student plays football, then the student is an athlete.

If a student plays soccer, then the student is an athlete.

Therefore, if a student plays football, then the student plays soccer.

14. If a shape is a rectangle, then it has four right angles.

If a shape is a square, then it has four right angles.

Therefore, if a shape is a rectangle, then it is a square.

15. If a shape is a square, then it has four sides.

If a shape has four sides, then it is a quadrilateral.

Therefore, if a shape is a square, then it is a quadrilateral.

16. If a person enjoys playing the guitar, then that person likes music.

If a person likes music, then that person can appreciate all genres of music.

Therefore, if a person enjoys playing the guitar, then that person can appreciate all genres of music.

17. If a number is divisible by 6, then it is divisible by 3.

If a number is divisible by 9, then it is divisible by 3.

Therefore, if a number is divisible by 6, then it is divisible by 9.

18. Some vehicles are cars.

A bus is a vehicle.

Therefore, a bus has four wheels.

19. Some fruit is red.

A banana is a fruit.

Therefore, a banana is red.

20. If a vehicle is a car, then it has four wheels.

If a vehicle is a truck, then it has four wheels.

Therefore, all cars are trucks.



Identify the flaw in the argument.

21. Bobby thought that there was a football game tonight. Bobby sees that his friend Jason, who is a member of the football team, is not wearing his jersey. Bobby says, “Since Jason isn’t wearing his jersey, then tonight’s game must be canceled.”

22. Stanley’s dad is the soccer coach for his little brother’s soccer team. Stanley asked his dad who won the game. His dad replied, “We outscored them 4 to 3 in the second half,” so Stanley congratulates his little brother on getting a win.

23. Keri knows that her mother is buying her a new car for her birthday. When Keri goes outside on her birthday, she sees a new red car on the street. Keri runs to her mother to thank her for the car.

24. Danitza needs to know the length of a string of spaghetti. Since her father told her that they were having spaghetti for dinner tonight, Danitza will wait for dinner and measure one that she gets on her plate.

25. The cover of Stewart’s book has four right angles. Stewart thinks his book’s shape must be a square.

26. Malika learned that every member of the cheerleading squad from last year is on this year’s squad as well. When she sees a girl with a cheerleading uniform on, she says to herself, “She must have been on the squad last year.”

27. Kara knows that an equilateral triangle has three 60° angles. Kara measures one angle of a triangle and the measure is 60°. Kara says the triangle must be an equilateral triangle.

28. Jessica needs to construct a triangular base for her table. She takes the fi rst three pieces that she sees in her dad’s workshop because you can make a triangle out of any three sizes of wood.





Hypothesis and Conclusion Conditional statements are the foundation of logical reasoning.

Notice in the defi nition that a statement is conditional if it can be expressed in if-then form. Not all conditional statements use the words if and then.

THINK ABOUT IT

In a conditional statement, the part that follows if is the hypothesis and the part that follows then is the conclusion. Sometimes the word then is not writ-ten, but is implied. It is also possible for a conditional statement to include a phrase that is neither the hypothesis nor the conclusion.

Identifying the Hypothesis and Conclusion of a Conditional StatementExample 1 Identify the hypothesis and conclusion of each conditional statement.A. If it is Wednesday, then I have a piano lesson.Solution The words following if are the hypothesis. The words following then are the conclusion.Hypothesis: it is WednesdayConclusion: I have a piano lesson ■B. A number is divisible by 2 if it is divisible by 4. Solution In this statement, the conclusion is written before the hypothesis.Hypothesis: it is divisible by 4Conclusion: a number is divisible by 2 ■C. For a ≥ 0, if x 2 = a, then x = ± √

__ a .

Solution In identifying the hypothesis and conclusion, disregard the premise, for a ≥ 0.Hypothesis: x 2 = a Conclusion: x = ± √

__ a ■

A conditional statement can also be called an implication, because the hypothesis implies the conclusion.

THINK ABOUT IT

The statement in Example 1B is equivalent to “If a number is divisible by 4, then it is divisible by 2.”

THINK ABOUT IT

A conditional statement is a statement that can be written in if-then form.

DEFINITION



Writing a Conditional StatementWhen writing a conditional in if-then form, it may be necessary to add extra words for clarity. Example 2 Write each statement as a conditional statement in if-then form. Then identify the hypothesis and conclusion.A. All cats like to eat salmon.Solution The statement implies that if an animal is a cat, then it would like to eat salmon. The conditional is:If an animal is a cat, then it likes to eat salmon.Hypothesis: an animal is a catConclusion: it likes to eat salmon ■B. Parallel lines have equal slopes.Solution If two or more lines are parallel, then their slopes are equal.Hypothesis: two or more lines are parallelConclusion: their slopes are equal ■

Identifying Conditional StatementsExample 3 Determine if each statement is a conditional statement. A. Shar will drive to the movie theater.Solution The statement does not give any condition for when Shar will drive to the movie theater, nor does it give any conclusion about what would happen if she does. The statement is not a conditional statement. ■B. Numbers that end in 5 are divisible by 5.Solution The statement implies that if a number ends in 5, then it is divis-ible by 5. The statement is a conditional. ■

Application: Real-World Conditional Statements Example 4

A. A merchant prints the words “All products come with a money-back guarantee” on each sales receipt. How can the merchant write the words as a conditional statement in if-then form?

Solution If you buy any product, then you get a money-back guarantee. ■B. One of the key recommendations for healthy living is “To maintain

body weight in a healthy range, balance calories from foods and bever-ages with calories expended.” How can you write the recommendation as a conditional statement in if-then form?

Solution If you balance calories from foods and beverages with calories expended, then you will maintain a body weight in a healthy range. ■

A conditional statement is sometimes just called a conditional.

TIP

HYPOTHESIS AND CONCLUSION 507




Problem Set

Identify the hypothesis and conclusion of each statement. 1. If it is Saturday, then I have a soccer game.

2. You can fi nd the decimal equivalent of a fraction if you divide the numerator by the denominator.

3. For all of values of a, b, and c, if a · b = a · c, then b = c.

4. If the temperature is below 32°F, then water will turn from a liquid to a solid.

5. If you were born in 2000, then you turned 5 in 2005.

6. If you live in Los Angeles, then you live in California.

7. If a word is a proper noun, then it starts with a capital letter.

8. If a triangle is a right triangle, then it has a 90° angle.

9. All photographers take pictures.

10. The sum of 12 and 14 is 26.

11. Every quadratic function has a graph that is shaped like a parabola.

12. Any number whose last two digits are divisible by 4 is divisible by 4.

13. The product of two negative numbers is a positive number.

14. All doctors must go to medical school.

15. Any student that cheats will fail.

16. I am going to walk my dog.

17. We can go to the pool after 10 a.m.

18. We don’t have school when it snows.

19. All mammals are warm-blooded.

20. Tasha wants to get her driver’s license.

21. Accidentally spilling bleach on your clothing will leave a stain.

22. Today is the last day of January.

Write each statement as a conditional statement in if-then form. Then identify the hypothesis and conclusion.

Determine if the statement is a conditional statement.

Rewrite each statement as a conditional statement.

23. The sum of the measures of the angles of a triangle is 180°.

24. A federal guideline regarding education states that students are required to pass tests in English, algebra, biology, and social studies before they can receive a diploma.

25. The local middle school publishes a student handbook outlining the rules of the school. One statement reads “Chronic lateness to school will result in suspension.”

26. State University has installed a fi lter to block school offi ces from accessing sports websites. The school issued a rule stating that employees who bypass the fi lter will be terminated.



Forms of Conditional Statements

You can form a new conditional statement from any conditional statement by switching the hypothesis and conclusion, negating the hypothesis and conclusion, or both switching and negating the hypothesis and conclusion.

Determining if a Converse Is True or False

FORMS OF CONDITIONAL STATEMENTS 509

When a conditional statement is true, the converse of that statement may or may not be true.Example 1 Write the converse of each conditional statement. Then deter-mine if the converse is true or false. A. If a number is an irrational number, then it is a real number.Solution The converse is: If a number is a real number, then it is an irra-tional number.The converse is false because real numbers are made up of both rational and irrational numbers. ■B. If ab = 0, then a = 0 or b = 0.Solution The converse is: If a = 0 or b = 0, then ab = 0.The converse is true. If any factor in a multiplication expression is 0, then the product is 0. ■C. All percents are ratios with denominators of 100.Solution It sometimes helps to first write the statement as a conditional statement: If an expression is a percent, then it is a ratio with a denominator of 100.The converse is: If an expression is a ratio with a denominator of 100, then it is a percent.The converse is true. This is the definition of a percent. ■

(continued)

The converse is a conditional statement that switches the hypothesis and conclusion of the original conditional statement.

Conditional: If a, then b.

Converse: If b, then a.

DEFINITION




D. An expression can be factored if it is a difference of squares.Solution Be careful: The conclusion is written before the hypothesis. Write the conditional in if-then form to see the hypothesis and conclusion.Conditional: If an expression is a difference of squares, then it can be factored.Converse: If an expression can be factored, then it is a difference of squares.The converse is false. Expressions other than differences of squares can be factored. ■

Writing Converses, Inverses, and ContrapositivesTo negate a statement means to write that the statement is not true. Negating hypotheses and conclusions gives us two more forms of conditional statements.

The inverse is a conditional statement that negates the hypothesis and the conclusion of the original conditional statement. The contrapositive is a conditional statement that both switches and negates the hypothesis and conclusion of the original conditional statement.

Conditional: If a, then b.

Converse: If b, then a.

Inverse: If not a, then not b.

Contrapositive: If not b, then not a.

DEFINITIONS

When a conditional statement is true, the inverse of that statement may or may not be true but the contrapositive will always be true. Example 2 Write the converse, inverse, and contrapositive of the conditional statement.Conditional: If it is raining, then I will go to the movies.Solution Identify the hypothesis and conclusion of the conditional.Hypothesis: it is rainingConclusion: I will go to the movies

Converse: If I go to the movies, then it is raining.Inverse: If it is not raining, then I will not go to the movies.Contrapositive: If I do not go to the movies, then it is not raining. ■

The contrapositive can be defi ned as the inverse of the converse.

THINK ABOUT IT



Determining if Converses, Inverses, and Contrapositives Are True or FalseExample 3 Determine if the converse, inverse, and contrapositive of the conditional statement is true or false.Conditional: All numbers that are divisible by 5 end in 0 or 5.Solution Write the statement in if-then form to identify the hypothesis and conclusion.If a number is divisible by 5, then it ends in 0 or 5.Hypothesis: A number is divisible by 5.Conclusion: It ends in 0 or 5.

Converse: If a number ends in 0 or 5, then it is divisible by 5.TrueInverse: If a number is not divisible by 5, then it does not end in 0 or 5.TrueContrapositive: If a number does not end in 0 or 5, then it is not divisible by 5. True ■

Using the Law of Contrapositives


If a conditional statement is true, then its contrapositive is also true.

THE LAW OF CONTRAPOSITIVES

The law of contrapositives can help you prove statements because sometimes it is easier or simpler to show the truth value of the contrapositive than the truth value of the conditional statement. Example 4 Use the law of contrapositives to determine whether the following statements are true.A. If a given triangle is not a right triangle, then a 2 + b 2 ≠ c 2 .Solution Identify the hypothesis and conclusion of the statement. Then write the contrapositive.Hypothesis: A given triangle is not a right triangle.Conclusion: a 2 + b 2 ≠ c 2

Contrapositive: If a 2 + b 2 = c 2 , then the triangle is a right triangle.The Pythagorean theorem guarantees that the contrapositive is true. So, the conditional statement is true. ■

(continued)

The truth value of a statement is either true or false.

TIP




B. If the discriminant is irrational, then the polynomial is not factorable.Hypothesis: the discriminant is irrationalConclusion: the polynomial is not factorable

Contrapositive: If the polynomial is factorable, then the discriminant is not irrational.The contrapositive is true since a polynomial is only factorable when the discriminant is a perfect square or 0. The contrapositive is true, so the original conditional is true ■

Application: Advertising Example 5 Alberto sees a billboard that reads, “Eat Green’s Granola Bars for a healthier you.” Alberto said, “Wow, if I don’t eat Green’s Granola Bars, then I won’t be healthy.” Explain the error in Alberto’s thinking.Solution The statement on the billboard can be written as the conditional, If you eat Green’s Granola Bars, then you will be healthy.Alberto’s statement is the inverse of the original statement and the inverse of a conditional statement is not always true. ■

Problem Set

Write the converse of each conditional statement. Then determine if the converse is true or false.

1. If a number is a rational number, then it can be written as a fraction.

2. If a function is a constant function, then the graph of the function is a horizontal line.

3. If a triangle is equilateral, then it is isosceles.

4. If n is prime, then it has n and 1 as factors.

5. All negative integers are less than 0.

6. All squares are rectangles.

7. If a polygon has three sides, then it is a triangle.

8. An expression cannot be factored if it is a sum of two squares.

Identify the hypothesis and conclusion. Then write the converse, inverse, and contrapositive of the conditional statement.

9. If it is Tuesday, then I have swimming lessons.

10. If I study, then I will pass the math test.

11. If I learn to cook, then I will stop eating at restaurants.

12. If I like this course, then I will be motivated to study.

13. If the football team wins the game this week, then they will be in the championship game.

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

15. If the dog gets sick, then I have to take him to the vet.

16. If I graduate from college, then I will have a college degree.



Identify the hypothesis and conclusion of the conditional statement. Then determine if the converse, inverse, and contrapositive of the conditional statement are true or false. 17. If a number is an integer, then it can be positive,

negative, or 0.

18. If an equation of a line is in slope-intercept form, then it is in the form y = mx + b.

19. All vertical lines have undefi ned slope.

20. All rational numbers are real numbers.

21. An expression is simplifi ed correctly if the order of operations is followed.

22. A year is a leap year if February has 29 days.

Identify the hypothesis and conclusion of the statement. Then write the contrapositive, and determine if the conditional statement and contrapositive are true or false.

23. If 5x ≠ 10, then x ≠ 2.

24. If an equation is quadratic, then it has two real roots.

25. If the discriminant is 0, then the graph of the quadratic function has one x-intercept.

26. If a function is linear, then its graph is a line.

*27. Challenge All quadratic variation equations have a constant of variation.

*28. Challenge If a number is a perfect number, then it is equal to half of the sum of all of its positive divisors.





Inductive and Deductive ReasoningInductive and deductive reasoning are commonly used to reach a conclusion.

Using Inductive Reasoning

Inductive reasoning is a type of reasoning based on observations of patterns or past events to reach a conclusion.

DEFINITION

Inductive reasoning does not always result in a true conclusion and it is not considered “proof” in mathematics. However, it does have an important place in mathematics. Example 1 Use inductive reasoning to complete each problem.A. Find the next term in the sequence 5, 15, 45, 135, 405, . . . .Solution Each term is 3 times the preceding term. Based on this pattern, the next term is 405 · 3 = 1215. ■B. Find the next term in the sequence 40, 39, 37, 34, 30, . . . .Solution Each term decreases by one more than the preceding term.40, 39, 37, 34, 30 −1 −2 −3 −4Based on this pattern, the next term is 30 − 5 = 25. ■C. Every Tuesday since he moved to town, Juan’s neighbor mows his lawn.

Today is Tuesday. Draw a conclusion. Solution Based on past events, Juan’s neighbor will mow his lawn today. ■

We are assuming in Example 1A that the pattern remains “multiply by 3,” but it could be “multiply by 3 four times, then add 1.” Our conclusion is an educated guess only.

THINK ABOUT IT

You can use inductive reasoning to generate new ideas to test deductively.

THINK ABOUT IT



Using Deductive Reasoning

Deductive reasoning is a type of reasoning that uses proven or accepted properties to reach a conclusion.

DEFINITION

Deductive reasoning uses logical reasoning, such as syllogisms, to move from one statement to another. Mathematical proofs use deductive reasoning.Example 2 Use deductive reasoning to draw a conclusion from the given statements.A. If a number ends in 0, then it is divisible by 10.

If a number is divisible by 10, then it is divisible by 5.Solution The statements are premises of a syllogism. The conclusion is, “If a number ends in 0, then it is divisible by 5.” ■B. If you study, then you will pass the test.

Daisy studied for the test.Solution Since anyone who studies will pass the test, Daisy will pass the test. ■C. If a number is prime, then it is not divisible by 2.

The number 1032 is divisible by 2.Solution The second statement is the negation of the conclusion of the first statement. A valid conclusion can be made by using the contrapositive of the first statement: “If a number is divisible by 2, then it is not a prime number.” Since we are given that the number 1032 is divisible by 2, we can conclude that it is not a prime number. ■The solution to Example 2C uses the Law of Detachment.

A syllogism is a type of argument that uses exactly three statements.

REMEMBER

INDUCTIVE AND DEDUCTIVE REASONING 515

Law of Detachment is a valid argument of the form:

If a, then b. a. Therefore, b.

DEFINITION




Identifying Inductive and Deductive ReasoningExample 3 Determine whether inductive or deductive reasoning was used.A. Julio wrote the following.

2 · 2 = 4 4 · 8 = 32 8 · 6 = 48

10 · 10 = 100

The product of any two even integers is an

even integer.

Solution Julio used a set of observations to create a conjecture. This is inductive reasoning. ■B. Lisa said that all equations have an equals sign, and all identities are

equations, so all identities must have equals signs.Solution Lisa used logical reasoning. Specifically, she used a form of syllogism. This is deductive reasoning. ■Example 4 Tyrell and Lee are each trying to prove that the product of a number plus 1 and the same number minus 1 equals 1 less then the square of the number.Tyrell wrote:

(5 + 1)(5 − 1) = 6 · 4 = 24 = 5 2 − 1 (0 + 1)(0 − 1) = 1 · (−1) = −1 = 0 2 − 1(−6 + 1)(−6 − 1) = (−5) · (−7) = 35 = (−6) 2 − 1 It is true for a positive number, zero, and a negative number, so it is true for all numbers.

Lee wrote:

(n + 1)(n − 1) = n 2 − n + n − 1 = n 2 − 1 by the FOIL method of polynomial multiplication.

Whose reasoning is considered mathematically sound proof?Solution Tyrell gave three examples while Lee used an accepted property of mathematics. Lee’s reasoning is deductive. Since deductive reasoning is accepted as proof in mathematics, Lee’s proof is mathematically sound. ■



Problem Set

Use inductive reasoning to complete each problem.

1. Find the next term in the sequence 1, 3, 5, 7, 9, . . . .


3. Every Friday Mr. Juarez gives a math quiz. Today is Friday. Draw a conclusion.

4. Find the next term in the sequence −5, −2, 1, 4, 7, . . . .

5. Find the next term in the sequence 250, 50, 10,

2, 2 __ 5 , . . . .

6. When it is not raining, Sami walks his dog. It is not raining today. Draw a conclusion.

7. The cafeteria closes the snack line on Thursday. Today is Thursday. Draw a conclusion.




Use deductive reasoning to draw a conclusion from the given statements.

11. If 3x − 2 = 6, then 3x = 8. If 3x = 8, then x = 8 __ 3 .

12. If a number is even, then it is divisible by 2. g = 14.

13. If the slope of a line equals 4, then the slope is positive. If the slope is positive, then the graph of the line rises from left to right.

14. If a number is divisible by 6, then it is also divisible by 3. The number 104 is not divisible by 3.

15. If the study session began at 4 p.m., then I am late. If I am late, then I will miss the discussion.

16. If a triangle is a right triangle, then it has one right angle. Triangle ABC does not have a right angle.

17. If it snows today, then the team practices indoors. The team is not practicing indoors today.

18. If a polygon has four sides, then it is a quadrilateral. A triangle is not a quadrilateral.

Determine whether inductive or deductive reasoning was used.

19. Joaquin wrote the following statements:

−1 · 2 = −2 −3 · 5 = −15

6 · (−2) = −12 4 · (−2) = −8

A negative number times a

positive number equals a

negative number.

20. Su Young said that all integers are rational numbers, and all rational numbers are real numbers, so all integers are real numbers.

21. Jon saw the following sequence: 2, 4, 8, 16, 32, . . . . He concluded that the sequence represented the powers of 2 and wrote the next three numbers: 64, 128, and 256.

22. Micah examined the following sequence: O, T, T, F, . . . . He concluded that the sequence represented the numbers One, Two, Three, Four, . . . and wrote the next four terms F, S, S, E, . . . .

23. If a number is a perfect square, then its square root is rational. The square root of 19 is not rational. Therefore, if a number’s square root is not rational, then the number is not a perfect square.

INDUCTIVE AND DEDUCTIVE REASONING 517




24. Sang wrote the following:

−1 · (−1) = 1 −2 · (−1) = 2

3 · (−1) = −3 10 · (−1) = −10

Multiplying a number by −1 gives

the opposite of the number.

25. Amara noticed that x = 2 in the second problem of her quiz. The value of x was also 2 in the fourth and sixth problems. Therefore, Amara reasoned that the answer to the eighth problem was x = 2.

26. If P = 2l + 2w, then 2l = P − 2w.

If 2l = P − 2w, then l = P − 2w _______ 2 . Therefore, if

P = 2l + 2w, then l = P − 2w _______ 2 .

27. Jamal’s best friend has gone to the library after school on Wednesday every week for the past month. Today is Wednesday. Jamal concludes that his friend will go to the library after school.

Solve.

28. Caleb and Thea are trying to prove that the square of a binomial equals a trinomial in the form of the sum of the square of the fi rst term, twice the product of the fi rst and second terms, and the square of the second term. Whose reasoning is considered a mathematically sound proof?

Caleb’s method:

(x + y) 2 = (x + y)(x + y) = x 2 + xy + xy + y 2 = x 2 + 2xy + y 2 by the FOIL method. This is true for all values of x and y.

Thea’s method:

(1 + 0) 2 = 1 2 + 2 · 1 · 0 + 0 2 = 1

(1 + 2) 2 = 1 2 + 2 · 1 · 2 + 2 2 = 1 + 4 + 4 = 9



ADDITION AND SUBTRACTION EQUATIONS 519

*29. Challenge Ed and Jeb are trying to prove that the factored form of x 3 + y 3 = (x + y)( x 2 − xy + y 2 ). Whose reasoning is considered a mathematically sound proof?

Jeb’s method:

( 2 3 + 3 3 ) = (2 + 3)( 2 2 − 2 · 3 + 3 2 )

= 5 · (4 − 6 + 9)

= 5 · 7 = 35

( 5 3 + 0 3 ) = (5 + 0)( 5 2 − 5 · 0 + 0 2 ) = 5 · 25 = 125

Ed’s method: Using the distributive property and the laws of multiplication,

(x + y)( x 2 − xy + y 2 )

= x · x 2 + x · (−xy) + x · y 2 + y · x 2

+ y(−xy) + y · y 2

= x 3 − x 2 y + x y 2 + x 2 y − x y 2 + y 3 = x 3 + y 3

This is true for all values of x and y.




*30. Challenge Jill and Tyne are trying to prove that the product of the slopes of two perpendicular lines equals −1. Whose reasoning is considered a mathematically sound proof?

Jill’s method:

m = 3 and m = − 1 __ 3 3 · ( − 1 __ 3 ) = − 3 __ 3 = −1

m = 3 __ 5 and m = − 5 __ 3 3 __ 5 · ( − 5 __ 3 ) = − 15 ___ 15 = −1

Tyne’s method:

The slopes of two perpendicular lines are opposite

reciprocals. Suppose that the fi rst line’s slope is a __ b and

the line perpendicular to the fi rst line has a slope of

− b __ a where a and b do not equal 0.

a __ b · ( − b __ a ) = − ab ___ ba = −1.

Therefore, the product of the slopes of two perpendicular lines equals −1.



Analyzing and Writing Proofs

You use deductive reasoning in proofs.

A proof is a clear, logical structure of reasoning that begins from accepted ideas and proceeds through logic to reach a conclusion. You will see most proofs written either in a two-column format or a paragraph format.

Justifying Steps in an Algebraic Proof Regardless of the format, each step in an algebraic proof must be justified. The justification can be a mathematical property, such as the associative property of multiplication or a definition, such as the definition of a rational number.

Justification can also come from postulates and theorems. Postulates are mathematical statements that have not been proven but are understood to be true. Some people say that postulates are self-evident. One well-known pos-tulate is that the whole is greater than the part. Unlike postulates, theorems are statements that have been or are to be proven on the basis of established definitions and properties.

When proving a conditional statement, the hypothesis is the given state-ment. The last line in the proof is the conclusion of the conditional statement.Example 1 Justify each step of the proof.Conjecture: If 2 x 2 + 6 = 22, then x = ±2 √

__ 2

2 x 2 + 6 = 22 2 x 2 = 16 x 2 = 8 x = ± √

__ 8

x = ± √____

4 · 2 x = ±2 √

__ 2

Solution

2 x 2 + 6 = 22 Given 2 x 2 = 16 Subtraction Property of Equality x 2 = 8 Division Property of Equality x = ± √

__ 8 Square Root Property

x = ± √____

4 · 2 Product Rule for Radicals x = ±2 √

__ 2 Simplify.

QED ■

Postulates are sometimes called axioms.

TIP

QED is an abbreviation for the Latin phrase “quod erat demonstrandum.” Some mathematicians use QED at the end of an argument to show that the proof is complete.

TIP

ANALYZING AND WRITING PROOFS 521




Identifying the Given in an Algebraic ProofExample 2 Determine the given statement in the proof of each conditional.A. y = 0 if 5y = 0.Solution The given statement is the hypothesis of the conditional, that is, the words following if. Therefore, 5y = 0 is the given statement. ■B. All whole numbers are rational numbers.Solution Write the statement in if-then form: If a number is a whole num-ber, then the number is a rational number. The given statement would be: x is a whole number. ■

Proving a Conjecture Although mathematicians use deductive reasoning to prove results, they use inductive reasoning to find conjectures to prove. For example, suppose a student writes the following equations.

3 + 3 = 6 7 + 11 = 18 13 + 1 = 1433 + 9 = 42 1 + 1 = 2 13 + 3 = 16

The student observes that all the addends are odd and all the sums are even and makes the following conjecture: the sum of two odd numbers is an even number. A conjecture is an educated guess. While the student found the conjecture by using inductive reasoning, she can only prove it with deductive reasoning. Example 3 Prove the conjecture: the sum of two odd numbers is an even number.Solution For any integer n, 2n is an even number. Because adding 1 to any even number results in an odd number, any odd number can be expressed in the form 2n + 1.Let 2a + 1 and 2b + 1 be any two odd numbers, where a and b are integers. Then their sum is 2a + 1 + 2b + 1.2a + 1 + 2b + 1 = 2a + 2b + 1 + 1 Commutative Property

of Addition = 2a + 2b + 2 Combine like terms. = 2(a + b + 1) Distributive Property

of MultiplicationBy the closure property of integers, a + b + 1 is an integer. Since 2 times any integer is an even number, 2(a + b + 1) is an even number. Therefore, the sum of two odd numbers is an even number. ■

Listing more sums of odd numbers does not prove the conjecture. There are infi nite pairs of numbers, and you can’t test all of them.

THINK ABOUT IT



Identifying the Flaw in a Mathematical ArgumentExample 4 Find the flaw in each proof.A. Given that a = 1, prove that 1 = 0. a = 1 Given a 2 = a Multiplication Property of Equality a 2 − a = 0 Subtraction Property of Equality

a 2 − a ______ a − 1 = 0 _____ a − 1 Division Property of Equality

a(a − 1)

________ a − 1 = 0 _____ a − 1 Factor.

a = 0 Simplify. 1 = 0 Substitution Property of EqualitySolution The error is in the fourth line when the division property of equality is used. Since a = 1, a − 1 = 0, and division by 0 is undefined. ■B. 3 = √

__ 9 = √

__________ (−3) · (−3) = √

_____ (−3) 2 = −3

Solution The product rule for radicals only applies when the factors are nonnegative. ■

Problem Set

Justify each step of the proof.

1. Conjecture: If 2(x + 6) = 10, then x = −1.

2(x + 6) = 10 A.

2x + 12 = 10 B.

2x = −2 C.

x = −1 D.

QED

2. Conjecture: If 3 x 2 + 10 = 100, then x = ± √___

30 .

3 x 2 + 10 = 100 A.

3 x 2 = 90 B.

x 2 = 30 C.

x = ± √___

30 D.

QED





3. Conjecture: If x 2 + 13x + 42 = 0, then x = −6 or x = −7.

x 2 + 13x + 42 = 0 A.

(x + 6)(x + 7) = 0 B.

x + 6 = 0 or x + 7 = 0 C.

x = −6 or x = −7 D.

QED

4. Conjecture: If 2 __ 3 x 2 − 40 = 0, then x = ±2 √___

15 .

2 __ 3 x 2 − 40 = 0 A.

2 __ 3 x 2 = 40 B.

3 __ 2 · 2 __ 3 x 2 = 3 __ 2 · 40 C.

x 2 = 60 D.

x = ± √___

60 E.

x = ± √_____

4 · 15 F.

x = ±2 √___

15 G.

QED

5. Conjecture: If 4( x 2 + 7) = 100, then x = ±3 √__

2 .

4( x 2 + 7) = 100 A.

4 x 2 + 28 = 100 B.

4 x 2 = 72 C.

x 2 = 18 D.

x = ± √___

18 E.

x = ± √____

9 · 2 F.

x = ±3 √__

2 G.

QED

6. Conjecture: If y = mx + b, then m = y − b

_____ x .

y = mx + b A.

mx + b = y B.

mx = y − b C.

m = y − b

_____ x D.

QED

Write the given statement you would use when proving the conditional.

7. x = ±2 if x 2 = 4.

8. If 3x + 5 = 20, then x = 5.

9. A statement is true if its contrapositive is true.

10. All linear functions have lines as graphs.

11. An equation has three roots, if it is a cubic function.

12. If the converse of a statement is true, then the inverse is also true.

Prove each conjecture.

13. The product of two even numbers is an even number.

14. The sum of an odd and even number is an odd number.

15. If a __ b = c __ d , then a + b _____ b = c + d _____ d .

16. The square of an odd is an odd number.

17. If a __ b = c __ d , then ad = bc.

18. If a __ b = c __ d , then a __ c = b __ d .



Find the flaw in each proof.

19. − 2 2 + 5 = 4 + 5 = 9

20. (−1) 100 = (−1) 99 · (−1) 1 = 1 · −1 = −1


21. Prove: If x 2 = x 3 , then x = 0, 1, or −1

x 2 = x 3 Given

x 2 − x 3 = 0 Subtraction Property of Equality

x(1 − x 2 ) = 0 Distributive Property of Multiplication

x(1 − x)(1 + x) = 0 Factoring a Difference of Two Squares

x = 0 or 1 − x = 0 or 1 + x = 0 Zero Product Property

x = 0 or x = 1 or x = −1 Addition and Subtraction Properties of Equality

22. Prove (x + 2) 2 = x 2 − 4

(x + 2) 2 = x 2 − 4 Given

= (x + 2)(x − 2) Laws of Exponents

= x 2 + 2x − 2x − 4 FOIL

= x 2 − 4 Additive Inverse Property

*23. Challenge The sum of the integers from 1 to 50 is 1275. Pairing the integers,

1 + 50 = 51

2 + 49 = 51

3 + 48 = 51

4 + 47 = 51

. . . There are 24 pairs so 24 · 51 = 1275. Therefore,

the sum of the integers from 1 to 50 is 1275.

*24. Challenge |−3| = −3

The defi nition of absolute value is

|x| = x if x ≥ 0−x if x < 0{ . Applying the defi nition,

|2| = 2 since 2 ≥ 0

|0| = 0 since 0 ≥ 0

|−3| = −3 since −3 < 0




It takes only one example to prove that a statement is false.

Counterexample

A counterexample is an example that shows a conjecture, statement, or theory is not a valid generalization.

DEFINITION

Finding Counterexamples for ConjecturesA conjecture can either be true or false. Deductive reasoning can prove a conjecture true. A single counterexample can prove a conjecture false. Note however, that just because nobody has found a counterexample does not mean a conjecture is proven. Example 1 Find a counterexample that disproves each conjecture.A. A student observes that 3, 5, 7, and 11 are all prime numbers that are also

odd. The student conjectures: All odd numbers are prime numbers. Solution Find an odd number that is not prime. One possible counterex-ample is 9. The number 9 is not prime because 3 · 3 = 9. ■

B. A student observes the following: (−3) 2 > 3, ( − 1 __ 2 ) 2 > − 1 __ 2 , and 10 2 > 10.

The student then conjectures: The square of any number is greater than the number.

Solution When trying to find a counterexample, try the values 0, 1, and fractions between 0 and 1. One possible counterexample is 0: 0 2 = 0. ■C. It is given that the points (0, 0) and (1, 1) are both contained in the graph

of a function. A student conjectures that the graph must be the graph of the function y = x.

A conjecture is a guess. It is often a conclusion of inductive reasoning.

REMEMBER

It is sometimes easier to prove a statement false than it is to prove a statement true.

TIP

For Example 1C, any number between 0 and 1 would also work.

TIP



COUNTEREXAMPLE 527

Solution There are other functions that contain the points (0, 0) and (1, 1). Two possible counterexamples are shown below.

4 521

(1,�1)

y�=�x2

(0,�0)3–2 –1–4–5 –3

5

4

3

2

1

–1

–2

–3

–4

–5

x

y

4 521

(1,�1)

y�=��x�

(0,�0)3–2 –1–4–5 –3

5

4

3

2

1

–1

–2

–3

–4

–5

x

y

■

Finding Counterexamples for ConditionalsYou can also prove that a conditional statement is false using a counterexam-ple. When finding a counterexample for a conditional, look for an example that makes the hypothesis true and the conclusion false. Example 2 Find a counterexample that disproves each conditional statement.A. If x 2 = 100, then x = 10.

Solution −10 is a counterexample because (−10) 2 = 100 and −10 ≠ 10. ■B. If a number is divisible by 3, then it is divisible by 6. Solution The number 3 is a counterexample because 3 is divisible by 3 but 3 is not divisible by 6. ■C. All prime numbers are odd numbers. Solution Write the conditional in if-then form: If a number is a prime num-ber, then it is an odd number. A counterexample is the number 2 because 2 is both prime and even. ■

Determining if a Counterexample Is ValidExample 3 Determine if each counterexample disproves the conjecture. If not, provide a counterexample that does.A. Conjecture: All parabolas intersect the x-axis. Counterexample:

4 521 3–2 –1–4–5 –3

5

4

3

2

1

–1

–2

–3

–4

–5

x

y




Solution No, a counterexample must be a parabola that does not intersect the x-axis.

4 521 3–2 –1–4–5 –3

5

4

3

2

1

–1

–2

–3

–4

–5

x

y

■B. Conjecture: If a polynomial can be factored, then its terms share a GCF

other than 1. Counterexample: x 2 − 4Solution Yes, the counterexample disproves the conjecture because it can be factored by a difference of squares but the terms do not have a GCF other than 1. ■

Problem Set

Find a counterexample that disproves each conjecture.

1. A student observes that 1, 2, and 3 are integers and positive numbers. The student conjectures: All integers are positive.

2. A student observes the following: 3 3 = 27, (−2) 3 = −8, 4 3 = 64, and 5 3 = 125. The student conjectures: Cubing an integer results in a multiple of that integer.

3. If 3 x 2 = 27, then x = −3.

4. It is given that the points (2, 4) and (−2, 4) are both contained in the graph of a function. A student conjectures that the graph must be the graph of the function y = 4.


6. If x 2 − 4 = 0, then x = 2.

7. A student compares the following values:

1 1 __ 1

2 1 __ 2

3 1 __ 3

The student conjectures: For all integers x, 1 __ x ≤ x.


9. It is given that the points (0, 0) and (4, 2) are both contained in the graph of a quadratic function. A student conjectures that the graph must be the graph of a parabola opening up.



COUNTEREXAMPLE 529

Determine if each counterexample disproves the conjecture. If not, provide a counterexample that does.

17. All triangles are right triangles.

Counterexample: an equilateral triangle

18. All equations containing an x with a power (or implied power) of 1 are examples of direct variation equations.

Counterexample: y = 14x

19. All lines have positive slope.

Counterexample: y = −3x + 5

20. The set of irrational numbers is closed under division.

Counterexample: √___

14 ____ √__

7 = √__

2

21. All rectangles are squares.

Counterexample:

22. All linear systems of equations have one solution.

Counterexample: 2x + 3y = 98x + 12 = 36

23. If you are given a slope and a point, you can always use the point-slope form to write an equation of the line.

Counterexample: m = 2, (−1, 5)

*24. Challenge All quadratic functions have two real roots.

Counterexample:

8 1042 6–4 –2–8–10 –6

10

8

6

4

2

–2

–4

–6

–8

–10

x

y

*25. Challenge All absolute value functions open upward.

Counterexample:

8 1042 6–4 –2–8–10 –6

10

8

6

4

2

–2

–4

–6

–8

–10

x

y

10. All rational numbers are in the form p __ q where

q ≠ 0.

11. A student notices that √__

2 and √__

3 are square roots and irrational numbers. The student conjectures: All square roots are irrational numbers.

12. All binomials can be factored.

13. A student observes that 4, 6, 8, and 16 are even and composite numbers. The student conjectures: All composite numbers are even.


15. A student compares the following values: 2 and 4, 3 and 9, and −4 and 16. The student conjectures: x 2 > x.

16. It is given that the points (0, 5) and (1, 5) are both contained in the graph of a function. A student conjectures that the graph must be the graph of a horizontal line.



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