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Exercises 3.5 Exercises

Exercise Group.
For the exercises below, list the set of integers that satisfy the given conditions.
1.
A positive multiple of 5 and not a multiple of 2
Answer.
{5, 15, 25, ...}
2.
Greater than 12 and less than or equal to 18
Exercise Group.
Write the negation of each quantified statement.
3.
Everyone failed the quiz today.
Answer.
At least one person did not fail the quiz today.
4.
Someone in the car needs to use the restroom.
5.
Some restaurants take reservations.
Answer.
No restaurants take reservations.
6.
All students drive to campus.
7.

Translate each statement from symbolic notation into English sentences. Let A represent “Elvis is alive” and let G represent “Elvis gained weight."

  1. \(\displaystyle A \vee G\)

  2. \(\displaystyle \sim \left( A \wedge G \right)\)

  3. \(\displaystyle G \to \sim A\)

  4. \(\displaystyle A \leftrightarrow \sim G\)

Answer.

  1. Elvis is alive or he gained weight.

  2. It is not true that Elvis is alive and gained weight.

  3. If Elvis gained weight, then he is not alive.

  4. Elvis is alive if and only if he did not gain weight.

8.

Translate each statement into words if \(d\) represents "dessert is included" and \(b\) represents "the bill was less than $20".

  1. \(\displaystyle d \wedge b\)

  2. \(\displaystyle \sim \left(d \vee b \right)\)

  3. \(\displaystyle \sim b \rightarrow d\)

9.

Let, p represent "I will attend the hockey game" and let q represent "I will enjoy the hockey game." Let r represent "I will watch a movie at home." Write each statement in symbolic form.

  1. I will attend the hockey game, but I will not enjoy it

  2. I’ll either attend the hockey game or watch a movie at home.

  3. It is not true that I will attend the hockey game and enjoy it.

Answer.

  1. \(\displaystyle p \wedge \sim q\)

  2. \(\displaystyle p \vee r \)

  3. \(\displaystyle \sim \left(p \wedge q \right)\)

10.

Let p represent "Rent is due on Friday" and q represent "We will go skiing."

Translate each statement in symbolic form.

  1. If rent isn't due on Friday, then we will go skiing.

  2. Rent is due on Friday and we will go skiing.

Exercise Group.
Create a truth table for each statement.
11.
\(A \wedge \sim B\)
Answer.
FTFF
12.
\(\sim \left( p \wedge q \right)\)
13.
\(\sim \left( \sim A \vee B \right)\)
Answer.
FTFF
14.
\(\left( r \wedge s \right) \vee \sim s\)
15.
\(\left( p \wedge \sim q \right) \wedge p\)
Answer.
FTFF
16.
\(\left( A \wedge B \right) \to \sim A\)
17.
\(p \vee \left( p \to q \right)\)
Answer.
TTTT
18.
\(\left( p \wedge q \right) \to p\)
19.
\(\left[ p \wedge \left( p \vee q \right) \right] \to p\)
Answer.
TTTT
20.

In this lesson, we have been studying the inclusive or, which allows both A and B to be true. The exclusive or does not allow both to be true; it translates to “either A or B, but not both.”

For each situation, decide whether the “or” is most likely exclusive or inclusive.

  1. An entrée at a restaurant includes soup or a salad.

  2. You should bring an umbrella or a raincoat with you.

  3. We can keep driving on I-5 or get on I-405 at the next exit.

  4. You should save this document on your computer or a flash drive.

21.

Translate each sentence into if-then form:

  1. All triangle are polygons.

  2. Every dog is happy.

22.

Consider the statement “If you are under age 17, then you cannot attend this movie.”

  1. Write the converse.

  2. Write the inverse.

  3. Write the contrapositive.

23.
Write the converse of "If you are older than 65, you will qualify for a discount."
Answer.
If you qualify for the discount, then you are older than 65.
24.
Write the inverse of "I will go Saturday if I get paid."
25.
Write the contrapositive of "If I don't receive my paycheck tomorrow, I won't pay you back."
26.

Assume that the statement “If you swear, then you will get your mouth washed out with soap” is true. Which of the following statements must also be true?

  1. If you don’t swear, then you won’t get your mouth washed out with soap.

  2. If you don’t get your mouth washed out with soap, then you didn’t swear.

  3. If you get your mouth washed out with soap, then you swore.

Exercise Group.
Write the negation of each conditional statement.
27.
If Luke faces Vader, then Obi-Wan cannot interfere.
Answer.
Luke faces Vader and Obi-Wan interferes.
28.
If you don’t look both ways before crossing the street, then you will get hit by a car.
29.
If you weren’t talking, then you wouldn’t have missed the instructions.
30.

Assume that the biconditional statement “You will play in the game if and only if you attend all practices this week” is true. Which of the following situations could happen?

  1. You attended all practices this week and didn’t play in the game.

  2. You didn’t attend all practices this week and played in the game.

  3. You didn’t attend all practices this week and didn’t play in the game.

Exercise Group.
Are the following examples of inductive or deductive reasoning?
31.
Erika brought cookies to class last Thursday, so she will bring cookies this Thursday.
32.
Roberto earned a 94% in math class. Any grade above a 90% gets an A. Therefore, Roberto will get an A in math class.
33.
All even numbers are divisible by 2. Eight is an even number. Therefore, 8 is divisble by 2.
34.
My sister didn't go home for Thanksgiving last year, so she won't go home for Thanksgiving this year.
35.
The roses in my backyard have bloomed every other year for the past 10 years. They didn't bloom last year, so they will bloom this year.
Exercise Group.
Use De Morgan’s Laws to rewrite each conjunction as a disjunction, or each disjunction as a conjunction.
36.
It is not true that Tina likes Sprite or 7-Up.
37.
It is not the case that you need a dated receipt and your credit card to return this item.
Answer.
You don’t need a dated receipt or you don’t need your credit card to return this item.
Exercise Group.
Use truth table to decide whether each argument is valid or invalid.
38.
All dogs go to heaven. Charlie is a dog. Therefore, Charlie will go to heaven.
39.
If Ronald eats pancakes before his test, he will ace it! Ronald aces his test. Therefore, he ate pancakes.
Answer.
Invalid
40.
If you like animals, you'll love my cat. You don't love my cat. Therefore, you don't like animals.
41.
If a person is on this reality show, they must be self-absorbed. Laura is not self-absorbed. Therefore, Laura cannot be on this reality show.
Answer.
Valid
42.
If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.
43.
Jamie must scrub the toilets or hose down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie will hose down the garbage cans.
Answer.
Valid
44.
Every student brought a pencil or a pen. Marcie brought a pencil. Therefore, Marcie did not bring a pen.
45.
If Aaron doesn't buy season tickets for the Seattle Kraken, he won't be able to go to the playoffs. Aaron went to the playoffs. Therefore, he bought season tickets.
Answer.
Valid
46.
If you miss the bus, you will be late for work. You didn't miss the bus. Therefore, you won't be late for work.
Exercise Group.
Write each argument in symbolic form. Then identify the form of the argument. Is it a valid or invalid argument?
47.
If the Vikings lose their next game, then the Packers will win. If the Packers win, they will go to the Super Bowl. Therefore, if the Vikings lose, the Packers will go to the Super Bowl.
Answer.
Valid; Transistive Property
48.
Sal has to work overtime or he won't be able to take Monday off. Sal was able to take Monday off. Therefore, he worked overtime.
49.
If I eat that piece of cake, then I will go to bed happy. I went to bed happy. Therefore, I ate that piece of cake.
50.
If Erik rents his house, then he can travel next month. Erik didn't rent his house. Therefore, he can't travel next month.
Exercise Group.
Name the type of logical fallacy being used.
51.
If you don’t want to drive from Boston to New York, then you will have to take the train.
Answer.
False dilemma; you could fly, take a bus, hitchhike…
52.
New England Patriots quarterback Tom Brady likes his footballs slightly underinflated. The “Cheatriots” have a history of bending or breaking the rules, so Brady must have told the equipment manager to make sure that the footballs were underinflated.
53.
Whenever our smoke detector beeps, my kids eat cereal for dinner.