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### Lesson 1: Mathematical Thinking and Logic

Lesson 1: Mathematical Thinking and Logic

Objectives

By the end of this lesson, students will be able to:

• decide whether information drawn from a passage is true, false, or cannot be determined
• recognize the hypothesis and conclusion of a conditional statement
• state conditionals in the if … then form
• recognize and state the converse, inverse, and contrapositive of a statement
• draw correct conclusions from given statements

Introduction

Geometry is a word taken from the Greeks with "geo" meaning earth and "metron" meaning measure. In about 300 BC, a Greek mathematician by the name of Euclid wrote a book called The Elements. It was a collection of all the known geometric facts organized in a logical manner. He included basic terms, like point, line, and plane, along with accepted principles called axioms or postulates. Euclid is known as the "Father of Geometry." In this lesson, you will apply critical thinking and decide whether information drawn from a passage is true, false, or cannot be determined. You will then learn how to recognize the hypothesis and conclusion of a conditional statement and how to state conditionals in the if… then form. Finally, you will learn to recognize and/or state the converse, inverse, and contrapositive of a statement, as well as draw correct conclusions from given statements.

Critical Thinking

The study of geometry requires one to be a critical thinker. Approaching and solving geometry problems is like a good detective analyzing a crime scene and looking for clues. You must be able to put your information in a clear, logical sequence to show how you arrived at your conclusion.

Example 1

Read the sentence below and determine if the statements that follow are true, false, or cannot be determined.

The Wilsons boarded a train to evacuate New Orleans two days prior to Hurricane Gustav coming ashore on Monday, September 1, 2008.

 Statement Explanation Gustav was a hurricane. True. The hurricane is named in the sentence. The Wilsons boarded the train on Friday. False. Two days before it came ashore would be Saturday, not Friday. The Wilsons' home is in New Orleans. Cannot be determined. The Wilsons could have been visiting the city, so they do not necessarily live there. The Wilsons have at least one child. Cannot be determined. The sentence does not mention how many people are in the Wilson family. The Wilsons left their car in New Orleans. Cannot be determined. Nothing was mentioned about a car.

As you can see from the example above, a valid conclusion cannot be drawn when you are missing a number of facts.

Example 2

Read the following passage and determine if the statements that follow are true, false, or cannot be determined.

The Wilsons arrived safely in Baltimore after boarding a train in New Orleans to escape Hurricane Gustav. Aunt Betty was waiting at the station two hours before they arrived. They hugged each other and headed for the parking garage. Soon after, they were in a car on the highway heading towards a good home-cooked meal.

 Statement Explanation The Wilsons left New Orleans to escape Hurricane Gustav. True. It is stated in the sentence. Aunt Betty lives in Baltimore. Cannot be determined. Aunt Betty could have driven from another city to Baltimore to pick up the Wilsons. The Wilsons arrived on time. Cannot be determined. The passage does not mention if the train was on time or not. The Wilsons boarded a train in Baltimore. False. Aunt Betty picked them up in her car. The train's final destination was Baltimore. Cannot be determined. The passage does not mention if the train was only making a stop in Baltimore.

Practice

Read the passage below and determine if the statements that follow are true, false, or cannot be determined.

The second before she opened her mouth, all Thomas could think was, "If I hear another horrible actor, I'm going to walk out that door." She wasn't, thankfully, but it didn't really matter, because he never would've walked out that door, the door in question being the one in the tiny rehearsal room at Harlequin Studios. It was a grungy-looking rehearsal studio, where everything smelled a little funny, like some undiscovered form of mildew. He wouldn't have walked out that door because auditions weren't over yet, and he was a professional, even though he wouldn't be getting paid. Neither would the actors--the more than 160 of them. The woman who was talking was actually pretty good, good enough to get called back. Her name, Skye Benrexi, was interesting enough alone to merit a callback.

Used with permission from David L. Williams

1. The rehearsal studio is located in New York City.

Cannot be determined Check Your Answer

2. There were more than 160 actors that came to the audition.

True Check Your Answer

3. Thomas was auditioning for a part.

False Check Your Answer

4. Each actor selected would receive a paycheck.

False Check Your Answer

5. Skye Benrexi will be selected for the part.

Cannot be determined Check Your Answer

6. The rehearsal room is located at Harlequin Studios.

True Check Your Answer

7. The rehearsal room was clean and fresh.

False Check Your Answer

8. Skye auditioned well.

True Check Your Answer

9. Thomas was a professional.

True Check Your Answer

10. Skye Benrexi is a stage name.

Cannot be determined Check Your Answer

Conditional Statements

Look at the following conditional statement :

The first clause begins with the word if - "if the Sun shines." This part of the statement is called the hypothesis. The second clause begins with the word then - "then you will see your shadow." This part of the statement is called the conclusion. The if statement is the given information, and the then statement is the conclusion or result of the given information. Conditional statements are used to establish certain conclusions.

In mathematics, we can label the if statement with the letter p and the then statement with a q. We can show the conditional statement using symbols in a few different ways:

if p, then q

p implies q

pq

Example 3

Identify the hypothesis and conclusion in the following statement:

If I score a 93% on the test, then I will have an A average.

hypothesis: I score a 93% on the test

conclusion: I will have an A average

Example 4

Identify the hypothesis and conclusion in the following statement:

If the cows are lying down in the field, then it will rain.

hypothesis: the cows are lying down in the field

conclusion: it will rain

Example 5

Identify the hypothesis and conclusion in the following statement:

If x and y are integers, then their product will be an integer.

hypothesis: x and y are integers

conclusion: their product will be an integer

Conditional statements are not always written in the if… then form.

Example 6

Identify the hypothesis and conclusion in the statement below. Rewrite the statement so it is in the form if… then.

When it rains, it pours.

hypothesis: it rains

conclusion: it pours

if… then form: If it rains, then it pours.

Example 7

Identify the hypothesis and conclusion in the statement below. Rewrite the statement so it is in the form if… then.

A banana is soft if it is spoiled.

hypothesis: a banana is spoiled

conclusion: it is soft

if… then form: If a banana is spoiled, then it is soft.

Practice

1-5. Identify the hypothesis and conclusion in the statements below. If a statement is not in the form if… then, rewrite the statement so it is in the form if… then.

1. If the lights are on in the house, then someone is home.

hypothesis: the lights are on in the house

conclusion: someone is home Check Your Answer

2. You should come inside if you get cold.

if… then form: If you get cold, then you should come inside.

hypothesis: you get cold

conclusion: you should come inside Check Your Answer

3. If a triangle has two equal sides, then it is an isosceles triangle.

hypothesis: a triangle has two equal sides

conclusion: it is an isosceles triangle Check Your Answer

4. The early bird gets the worm.

if… then form: If the bird is early, then it gets the worm.

hypothesis: the bird is early

conclusion: it gets the worm Check Your Answer

5. A straight angle measures 180 degrees.

if… then form: If an angle is a straight angle, then it measures 180 degrees.

hypothesis: an angle is a straight angle

conclusion: it has a measure of 180 degrees Check Your Answer

Equivalent Statements

The statement "A right angle has a measure of 90 degrees" can be written as a conditional statement:

If an angle is a right angle, then it has a measure of 90 degrees.

It is now in the form of "if p then q." If the p and q are reversed, the statement becomes:

If an angle has a measure of 90 degrees, then it is a right angle.

When the p and q are reversed, the converse is formed. The converse of pq is written qp. In this example, both the conditional statement and its converse are true. This is called a biconditional statement. A statement is biconditional if both it and its converse are true. This is not always the case. Look at another conditional statement:

If you live in Miami, then you live in Florida.

Reversing the two parts of the statement gives us the converse:

If you live in Florida, then you live in Miami.

In this case the converse is not necessarily true. A person can live in other cities besides Miami and still be in Florida.

If we want to negate or say the negative of a statement, we use the word not. Once again, look at the conditional statement "If an angle is a right angle, then it measures 90 degrees." To make the negative of this statement, add not to each side:

If an angle is not a right angle, then it does not measure 90 degrees.

We use the symbol ~ for the word not. In this case, we can use symbols to describe this statement as ~p → ~q.

When you negate both the hypothesis and the conclusion, you have written the inverse. In the example above, both the conditional and its inverse are true, but this is not always the case. Look at the statement about Florida again:

If you live in Miami, then you live in Florida.

The inverse of this statement is not necessarily true:

If you do not live in Miami, then you do not live in Florida.

We can also take the inverse of the converse of a statement.

Conditional statement: If an angle is a right angle, then it measures 90 degrees.

Converse: If an angle measures 90 degrees, then it is a right angle.

Take the inverse of the converse: If an angle does not measure 90 degrees, then it is not a right angle.

The inverse of a converse is called the contrapositive. For example, the contrapositive of the statement "If you live in Miami, then you live in Florida" is the following:

If you do not live in Florida, then you do not live in Miami.

In both cases above, note that the contrapositive is true. The contrapositive is always logically equivalent to the conditional statement (in other words, they both have the same meaning).

Here is a way to remember conditionals: Example 8

Accept the following conditional statement as true:

If you live in Winchester, then you attend St. Augustus High School.

Write the converse, inverse, and contrapositive of each statement. Say whether each is true, false, or not necessarily true (cannot be determined), and which if any are logically equivalent.

 Converse If you attend St. Augustus High School, then you live in Winchester. Explanation Not necessarily true. The statement does not indicate whether all students at St. Augustus High School live in Winchester. Inverse If you do not live in Winchester, then you do not attend St. Augustus High School. Explanation Not necessarily true. The statement does not indicate whether all students who do not attend St. Augustus High School do not live in Winchester. Contrapositive If you do not attend St. Augustus High School, then you do not live in Winchester. Explanation True. According to the converse, if you attend St. Augustus High School, then you live in Winchester. The conditional and the contrapositive are logically equivalent. If you live in Winchester, then you attend St Augustus High School, so if you do not attend St Augustus High school, then you do not live in Winchester.

Practice

1. Accept the following conditional statement as true:

If you are on Flight #107, then you are going to Atlanta.

Write the converse, inverse, and contrapositive of the statement, and say which statements, if any, are logically equivalent.

Converse: If you are going to Atlanta, then you are on Flight #107. (not necessarily true)
Inverse: If you are not on Flight #107, then you are not going to Atlanta. (not necessarily true)
Contrapositive: If you are not going to Atlanta, then you are not on Flight #107. (true)
The conditional and the contrapositive are logically equivalent. The converse and inverse are also logically equivalent. Check Your Answer

2. Identify the relation of each of the following statements to the original statement below:

If you use a boogie board, then you live along the coast.

a) If you do not live along the coast, then you do not use a boogie board.

b) If you live along the coast, then you use a boogie board.

c) If you do not use a boogie board, then you do not live along the coast.

a) contrapositive

b) converse

c) inverse Check Your Answer

Deductive Reasoning

In presenting a logical argument, one must combine at least three conditional statements in the following form (the symbol for "therefore" is ∴ ).

pq

qr

pr

Above, if the first two statements (premises) are true, then the third statement (conclusion) must also be true. Notice the hypothesis of one premise is the same as the conclusion of the other premise. This is called a syllogism. If the first and second premises are true, then the final conclusion is also true.

Example 9

If a student walks to school, he is late for school. If a student is late for school, he misses part of his first class. What conclusion can you reach based on these premises?

 p → q If a student walks to school (p), he is late for school (q). q → r If a student is late for school (q), he misses part of his first class (r). ∴ p → r Therefore, a student who walks to school misses part of his first class.

Example 10

If you live in Tampa, then you live in Hillsborough County. If you live in Hillsborough County, then you live in Florida. What conclusion can you reach based on these premises?

 p → q If you live in Tampa (p), then you live in Hillsborough County (q). q → r If you live in Hillsborough County (q), then you live in Florida (r). ∴ p → r Therefore, if you live in Tampa you live in Florida.

Practice

1. Louis lives in the Ninth Ward. The Ninth Ward is in New Orleans. What conclusion can be reached from these premises?

Therefore, Louis lives in New Orleans. Check Your Answer

2. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

pq

pr

No (The hypothesis of one is not the conclusion of the other.) Check Your Answer

3. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

qr

pq

Yes, pr (The order of the premises can be changed, so pq, qr, pr.) Check Your Answer

Multiple Premises

Sometimes there are more than two premises to consider.

pq

qr

rs

st

pt

This is what a good detective does when solving a crime. One clue will lead to another. Eventually, when all the pieces fit together in the proper order, the crime is solved. Look at the premises below that lead to a conclusion. Notice the conclusion of one statement is the hypothesis of the next statement.

If Alvin has \$80, then he will go shopping.

If Alvin goes shopping, then he will buy baseball shoes.

If Alvin buys baseball shoes, then he will try out for the team

If Alvin tries out for the team, then he will make the team.

Therefore, if Alvin has \$80, then he will make the baseball team.

Sometimes the steps to prove a syllogism are not as obvious as those in the preceding problems. An equivalent statement may need to be written in order to have the conclusion of one premise be the hypothesis of the other. Equivalent statements can be substituted for one another. Recall that the conditional and its contrapositive are equivalent. Look at the examples below of equivalent statements:

pq is equivalent to ~q → ~p

~pq is equivalent to ~qp

q → ~p is equivalent to p → ~q

Each of these could be substituted to produce a syllogism that leads to a conclusion.

Example 11

Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

c → ~d

ed

Use the fact that the contrapositive of c → ~d is d → ~c

Rewrite the pair of premises, changing the order and substituting the contrapositive, to find the conclusion.

ed

d → ~c

e → ~c

Practice

1. If Peter makes an A in his geometry class, he will spend the summer in the Caribbean. If he goes to the Caribbean, he gets to go to the summer festival. If he goes to the summer festival, he will perform with the band. What conclusion can be drawn from these premises?

Therefore, if Peter makes an A in his geometry class, he will perform with the band at the summer festival in the Caribbean. Check Your Answer

Truth Tables

Earlier in the lesson you learned how conditional, converse, inverse, and contrapositive statements can be equivalent. Now we are going to verify which ones are equivalent by using a truth table. A truth table shows all the possibilities in a given situation.

It is raining.

You get wet.

Both of these statements can either be true or false. Assign the letter p to the first statement: "it is raining." The second statement, "you get wet," will be the letter q. Now we will put these two statements together as a conditional statement.

If it is raining, then you get wet.

In symbolic form this is written:

pq

A truth table is a way to display all the possibilities about whether either of those statements are true. It shows the original, converse, inverse, and contrapositive. First look at a truth table for the two parts of the conditional statement above:

 p q True True True False False True False False

Notice that there are four possible combinations of true and false for p and q. Now look at the possibilities for the whole statement, pq. (From now on we will use T for true and F for false.) The third column gives what is called the truth value of a statement. The only time that pq is false is when p is true and q is false. In other words, a true first term, or premise, cannot imply a false conclusion.

 p q p → q T T T T F F F T T F F T

In the first case, T → T. If both of the statements--that it is raining and that you get wet --are true statements, then the conditional statement is true. So, the truth value of this is T.

The second case is T → F. This means that the first statement, it is raining, is true, but the second statement, that you get wet, is not. If this is the case, the conditional statement cannot be true. The truth value here is F, false.

In the third case, F → T. Here, the first statement is not true, so it is not raining. The second statement, you get wet, is true. In this case, the truth value is T, because we do not have enough information to know that it is false.

In the fourth case, F → F. In this case, the first and second statements are both false. As with the example above, since we do not have any information about what happens when it is not raining, the statement cannot be proven false. Here, the truth value is T.

Negation

The symbol for negation (or "not") is a tilde (~). If we want to show that a statement p has been negated, we write ~p. The truth table for negation is the following:

 p ~p T F F T

Negation causes a true statement to become false and a false one to become true. Say the original statement is "10 plus 5 equals 15." This is true. The negation of this statement is "10 plus 5 does not equal 15." This is false. In another example, the original statement is "A pig is a horse." This is false. The negation of this statement is "A pig is not a horse." This is true.

Conjunction

Another symbol that we use in truth tables is the conjunction or "and" symbol. If we want to connect the statements p and q with the word and, we write it as p q.

Look back at the example above. We will connect the statements with a conjunction:

It is raining and you get wet.

p q

The truth table for conjunctions is the following:

 p q p ∧ q T T T T F F F T F F F F

Look at an example. Check to see if the conjunction statement is true based on the truth table:

Green is a color and 4 + 5 = 9.

Look at the table. Both of these statements are true. Based on the first row of the table, the entire sentence is true.

Now look at a similar statement:

Green is a color and 4 + 5 = 10.

Since the first statement is true and the second is false, the entire sentence is false.

Example 12

Use the truth table above to state the truth value of the sentence below.

An apple is a vegetable and 18 - 5 = 13.

The first statement is false--an apple is not a vegetable.

The second statement is true.

Since the first statement is false and the second is true, the entire sentence is false.

Example 13

Use the truth table above to state the truth value of the sentence below.

An apple is a vegetable and 18 - 5 = 10.

The first statement is false.

The second statement is also false.

Since the first statement is false and the second is false, the entire sentence is false.

Notice that the only time p q is true is when both p and q are true.

Practice

1 - 4. Determine the truth value of the following sentences.

1. 5 + 7 = 12 and 5 is an even number.

The first statement is true, but second statement is false. Therefore, the whole sentence is false. Check Your Answer

2. London is the capital of England and Paris is the capital of France.

The first statement is true, and the second statement is true. Therefore, the whole sentence is true. Check Your Answer

3. Eleven is not a prime number and there are five sides in a pentagon.

The first statement is false, though the second statement is true. Therefore, the whole statement is false. Check Your Answer

4. The opposite sides of a trapezoid are always equal and there are 350° in a circle.

The first statement is false and the second statement is false, so the whole statement is false. Check Your Answer

Equivalent Statements

Look again at the statements we have been using:

p: It is raining.

q: You get wet.

We can use this statement and truth tables to show which statements are equivalent.

A. Conditional statement:

If it is raining, then you get wet.

pq

We have seen this truth table before.

 p q p → q T T T T F F F T T F F T

B. Converse statement:

If you get wet, then it is raining.

qp

Notice that the p and q have been switched. The truth table looks like the following:

 I II III p q q → p T T T T F T F T F F F T

We have labeled the columns with Roman numerals. In order to get column III, you must do column II → column I. Notice that the results are different from A.

C. Inverse statement:

If it is not raining, then you do not get wet.

~p → ~q

Because p and q have been negated, you have to put the negation columns in the truth table. Columns III and IV are used to form column V.

 I II III IV V p q ~p ~q ~p → ~q T T F F T T F F T T F T T F F F F T T T

D. Contrapositive statement:

If you do not get wet, then it is not raining.

~q → ~p

Notice that the p and q have been switched and negated. We must put in the negation columns. In order to get column V, you must do column IV → column III.

 I II III IV V p q ~p ~q ~q → ~p T T F F T T F F T F F T T F T F F T T T

Notice that the last column in the conditional (A) and the contrapositive (D) tables are the same. Therefore, we say that they are equivalent statements. Also, the last column in the converse (B) and the inverse (C) tables are the same. Therefore, they are equivalent.

Validity of an Argument

Truth tables can also be used to determine the validity of an argument. An argument consists of premises (hypothesis) and a conclusion. The premises are connected with the word "and" (shown with the symbol ).

Look at an example of an argument.

 Premises: If the rain stops, I go outside. The rain stops Conclusion: I go outside.

Write the statements with symbols.

p: The rain stops.
q: I go outside.

Put premises together with the symbol for a conjunction ().

(pq) p

The conclusion is q. The premises and conclusion are stated as an implied (→) statement.

[(pq) p] → q

Let's build a truth table to look at this statement.

 p q p → q p → q ∧ p [p → q ∧ p] → q

Notice that we start with p and q and build from there. The last column is the final statement. Now we will fill in the truth table, using the four possibilities.

 I II III IV V p q p → q p → q ∧ p [p → q ∧ p] → q T T T T T T F F F T F T T F T F F T F T

Column III is formed by using the rules for → (implied).

III = I → II

Column IV is formed by using the rules for (and).

IV = III I

Column V is formed by using the rules for → (implied).

V = IV → II

In the last column (V), the results are all true. When this happens it is called a tautology, and the argument is valid.

Look at another argument.

If it is raining, then I am reading.

Therefore, it is raining.

p: It is raining.

 Premises: p → q q Conclusion: p Argument: [p → q ∧ q] → p

Now set up the truth table:

 I II III IV V p q p → q p → q ∧ q [p → q ∧ q] → p T T T T T T F F F T F T T T F F F T F T

Column III is formed by using the rules for → (implied).

III = I → II

Column IV is formed by using the rules for (and).

IV = III II

Column V is formed by using the rules for → (implied).

V = IV → I

In the last column (V), the results are not all true. Therefore, the argument is not valid. Here is another argument.

Example 14

Construct and describe a truth table about the following argument.

If wool is expensive, then sweaters are expensive.

Sweaters are not expensive.

Therefore, wool is not expensive.

p: Wool is expensive.

q: Sweaters are expensive

 Premises: p → q ~q Conclusion: ~p Argument: [p → q ∧ ~q] → ~p

Notice that there are negations needed here. Add negation columns to the truth table.

 I II III IV V VI VII p q ~p ~q p → q p → q ∧ ~q [p → q ∧ ~q] → ~p T T F F T F T T F F T F F T F T T F T F T F F T T T T T

Columns III and IV are formed by negating Columns I and II.

Column V is formed by using the rules for → (implied).

V = I → II

Column VI is formed by using the rules for (and).

VI = V IV

Column VII is formed by using the rules for → (implied).

VII = VI → III

The last column is all true. Therefore, the argument is valid.

Practice

1. Construct and explain a truth table about the following argument.

If John does not go to town, then Marty stays home.

John goes to town.

Therefore, Marty does not stay home.

p: John goes to town.

q: Marty stays home.

 Premises: ~p → q p Argument: [(p → q) ∧ ~p] → ~q

 I II III IV V VI VII p q ~p ~q ~p → q (~p → q) ∧ p [(~p → q) ∧ p] → ~q T T F F T T F T F F T T T T F T T F T F T F F T T F F T

Columns III and IV are formed by negating Columns I and II

Column V is formed by using the rules for → (implied).

V = III → II

Column VI is formed by using the rules for (and).

VI = V I

Column VII is formed by using the rules for → (implied).

VII = VI → IV

The last column is not all true. Therefore, the argument is not valid. Check Your Answer

Enrichment Activity

Research the history of geometry. There are many resources that contain information on how geometry was first used by the Babylonians, Egyptians, Hindus, Chinese and Greeks. Famous mathematicians such as Euclid, Archimedes, Thales, and Pythagoras were instrumental in the development of geometry. Do some reading on these ancient times and mathematicians, and write a summary of what you have learned. One possible resource is a book titled Great Moments in Mathematics Before 1650 by Howard Eves, published by the Mathematical Association of America.

Lesson Review & Homework

We started this lesson by defining conditional statements, also known as if … then statements. A conditional statement is made up of a hypothesis and a conclusion. Then, we defined converse, inverse, and contrapositive statements. The lesson also covered which of these statements are logically equivalent--that is, they mean the same thing. The converse and inverse statements are equivalent. The conditional and contrapositive are equivalent.

You used this information when applying deductive reasoning and finding syllogisms, which are used to form a true conclusion from given premises. You learned about truth tables. You also learned how to use them to understand negation, conjunctions, and the validity of arguments. This kind of logical thinking will be the basis for much of the work you will do as you explore geometry.

Homework

1.

Identify the hypothesis and conclusion of each of the following conditional statements. Rewrite any statement that is not already in the if… then form.

a. If the geese are flying south, then winter is on its way.

b. If you study hard, then you will achieve success.

c. The diameter of a circle is 14 inches if the radius of the circle is 7 inches.

d. Those that live in Florida do not have to pay a state income tax.

e. A scalene triangle has no sides of equal length.

2.

Identify the relation of the following statements to the original statement below:

The grass will grow if it rains.

a. If it doesn't rain, the grass will not grow.

b. If the grass grows, then it rained.

3.

Write the converse of the statement below. Is it a good definition of what it means for an animal to be extinct? Why or why not?

If a type of animal is extinct, then there are no animals of that type left on Earth.

4.

Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

pq

rq

5.

Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

pq

rp

6.

Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.

xy

z → ~y

7.

Determine if the following pairs of premises form a syllogism. If they do, state the conclusion

~r → ~s

st

8.

Form the negation of each statement.

a. The sky is cloudy.

b. The Pittsburgh Steelers football team is not the team with the most Super Bowl wins.

c. Josie is the only person with red hair.

9.

Let p and q represent the following simple statements. Write each sentence below in symbolic form.

p: This is a cow.

q: This is a mammal.

a. If this is a cow, then this is a mammal.

b. If this is a mammal, then this is a cow.

c. If this is not a mammal, then this is not a cow.

10.

Let q and r represent the following simple statements. Write each symbolic statement below in words.

q: It is Memorial Day.

r: There will be fireworks.

a. q ~r

b. r → ~q

c. ~r ~q

d. ~q → ~r

11.

11. Use truth tables to show that qp and ~ p → ~ q are equivalent.

12.

By using a truth table, determine the validity of the following argument:

If we turn off the TV, there will be less noise.

There is less noise.

Therefore, we turned off the TV.