2.5: Review Exercises

1. Determine whether the following are statements for the purposes of logic. For each statement, tell whether it is simple or compound.

1. Shut the door.
2. Houston is a city in Texas.
3. There are 7 hours in a day and there are 24 days in a week.
4. \(2 + 3 = 6\)
5. What time is it?
6. If Darren passes the course, then he will graduate.

2. Translate each statement from symbolic notation into a word sentence, where \(a\) represents “Elvis is alive” and let \(g\) represents “Elvis gained weight.”

1. \(\sim g\)
2. \(a \vee g\)
3. \(\sim(a \wedge g)\)
4. \(g \rightarrow \; \sim a\)
5. \(a \leftrightarrow \; \sim g\)

3. Use simple statements \(p\), \(q\), and \(r\) to translate each word sentence into symbolic notation.

\(p\): "The lasagna is hot."        \(q\): "The breadsticks are cold."        \(r\): "The pizza will be delivered."

1. The pizza will not be delivered.
2. The lasagna is hot or the breadsticks are cold.
3. If the breadsticks are cold, then the pizza will be delivered.
4. If the pizza won't be delivered, then the lasagna is hot and the breadsticks are cold.
5. The breadsticks aren't cold if and only if the lasagna isn't hot.
6. It is not true that if the pizza is delivered, then the breadsticks are cold.

4. Use simple statements \(p\), \(q\), \(r\), and \(s\) to translate each symbolic statement into a word sentence. Then, determine the truth value of the resulting statement.

\(p\): The moon is made of cheese.      \(q\): We live on Earth.      \(r\): Earth has one moon.      \(s\): Earth is flat.

1. \(\sim p\)
2. \(p \vee q\)
3. \(q \wedge r\)
4. \(\sim(p \wedge r)\)
5. \(r \rightarrow p\)
6. \(p \rightarrow q\)
7. \(p \leftrightarrow s\)

5.  Complete the partial truth tables below to determine the truth value of the statements in the last column.

a.

\(\begin{array}{|c|c|c|c|c|c|c|}
\hline p & q & r & p \vee q & (\sim p \vee q) & \sim r & {\sim(p \vee q) \rightarrow \sim r} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{} & \mathrm{} & \mathrm{} & \mathrm{}\\
\hline
\end{array}\)

b.

\(\begin{array}{|c|c|c|c|c|c|c|}
\hline p & q & \sim q & p \rightarrow q & \sim (p \rightarrow q) & p \ \wedge \sim q & {\sim(p \rightarrow q) \leftrightarrow (p \ \wedge \sim q)} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{} & \mathrm{} & \mathrm{} & \mathrm{} & \mathrm{}\\ \hline
\end{array}\)

 

6. Create a complete truth table for each compound statement. Then tell whether the statement is a tautology, self-contradiction, or neither.

1. \(p \; \wedge \sim q\)
2. \(q \ \wedge \sim(\sim p \vee q)\)
3. \(p \rightarrow (\sim q \vee p)\)
4. \((\sim p \ \vee \sim q) \leftrightarrow q\)
5. \((p \ \vee q) \rightarrow \; \sim r\)

7. Use the order of logical operations to determine the truth value of each compound statement. 

1. \(\sim q \rightarrow  \sim p \vee q\)   when \(p\) is true and \(q\) is false.
2. \(\sim (p \ \wedge q) \leftrightarrow  \sim p \ \vee \sim r\)   when \(p\) is true, \(q\) is false, and \(r\) is false.

8. Use a truth table to determine whether the two statements are logically equivalent.

\(\sim (p \wedge q)\) and \(\sim p \; \wedge \sim q\)

9. Use one of DeMorgan’s Laws to rewrite each statement.

1. It is not true that Tina likes Sprite or 7-Up.
2. It is not the case that you need a dated receipt and your credit card to return this item.
3. I am not going or she is not going.

10. Use one of DeMorgan's Laws to select the statement that is equivalent to the negation of "Today is Monday and it isn't raining."

1. Today isn't Monday and it isn't raining.
2. Today isn't Monday or it isn't raining.
3. Today isn't Monday or it is raining.
4. Today isn't Monday and it is raining.

11. Rewrite each statement using the logical equivalency provided.

1. If you were talking, then you missed the instructions. Use \(p \rightarrow q \  \equiv \ \sim p \vee q \).
2. It is not true that if Luke faces Vader, then Obi-Wan cannot interfere. Use \(\sim(p \rightarrow q) \  \equiv \  p \ \wedge \sim q \).
3. If you don’t look both ways before crossing the street, then you will get hit by a car.  Use \(p \rightarrow q \  \equiv  \ \sim q \rightarrow  \sim p \).

12. Select the statement that is logically equivalent to "If we get a pay raise, then we will be happy."

1. We don't get a pay raise or we are happy.
2. We get a pay raise and we are happy.
3. We get a pay raise and we aren't happy.
4. We don't get a pay raise or we aren't happy.

13. Select the statement that is logically equivalent to the negation of "If you know the password, then you can get in."

1. You know the password and you can get in.
2. You don't know the password or you can get in.
3. You don't know the password and you can't get in.
4. You know the password and you can't get in.

14. Consider the conditional statement “If you read a newspaper, then you will learn something.”

1. Write its converse.
2. Write its inverse.
3. Write its contrapositive.

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

1. Write its converse.
2. Write its inverse.
3. Write its contrapositive.

 

16. Select the statement that is logically equivalent to “If you eat that day-old burrito, you will use lots of hot sauce.”

1. If you didn’t use lots of hot sauce, then you didn’t eat that day-old burrito.
2. If you don’t eat that day-old burrito, then you won’t use lots of hot sauce.
3. If you used lots of hot sauce, then you ate that day-old burrito.
4. All three statements are equivalent to the given statement.

17. Select the statement that is logically equivalent to "If I don't invest wisely, then I'll lose my money."

1. If I lose my money, then I didn't invest wisely.
2. If I don't lose my money, then I invested wisely.
3. If I invest wisely, then I won't lose my money.

18. Use a truth table to decide whether each argument is valid or invalid.

1. \(\begin{array} {ll} \text{Premise 1:} & p \vee q \\ \text{Premise 2:} & p \\ \text{Conclusion:} & \sim q \end{array}\)

 

2. \(\begin{array} {ll} \text{Premise 1:} & p \rightarrow q \\ \text{Premise 2:} & \sim q \\ \text{Conclusion:} & \sim p \end{array}\)

 

19.  Rewrite each argument symbolically. Then, determine whether the argument is valid or invalid.  

1. If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.
2. Jamie scrubs the toilets or hoses down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie hoses down the garbage cans.
3. If I don’t change my oil regularly, then my engine dies. My engine died. Therefore, I didn’t change my oil regularly.