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.
- Shut the door.
- Houston is a city in Texas.
- There are 7 hours in a day and there are 24 days in a week.
- \(2 + 3 = 6\)
- What time is it?
- 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. ”
- \(\sim g\)
- \(a \vee g\)
- \(\sim(a \wedge g)\)
- \(g \rightarrow \; \sim a\)
- \(a \leftrightarrow \; \sim g\)
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 ."
- The pizza will not be delivered.
- The lasagna is hot or the breadsticks are cold.
- If the breadsticks are cold, then the pizza will be delivered.
- If the pizza won't be delivered, then the lasagna is hot and the breadsticks are cold.
- The breadsticks aren't cold if and only if the lasagna isn't hot.
- 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.
- \(\sim p\)
- \(p \vee q\)
- \(q \wedge r\)
- \(\sim(p \wedge r)\)
- \(r \rightarrow p\)
- \(p \rightarrow q\)
- \(p \leftrightarrow s\)
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 .
- \(p \; \wedge \sim q\)
- \(q \ \wedge \sim(\sim p \vee q)\)
- \(p \rightarrow (\sim q \vee p)\)
- \((\sim p \ \vee \sim q) \leftrightarrow q\)
- \((p \ \vee q) \rightarrow \; \sim r\)
7. Use the order of logical operations to determine the truth value of each compound statement.
- \(\sim q \rightarrow \sim p \vee q\) when \(p\) is true and \(q\) is false.
- \(\sim (p \ \wedge q) \leftrightarrow \sim p \ \vee \sim r\) 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.
- It is not true that Tina likes Sprite or 7-Up.
- It is not the case that you need a dated receipt and your credit card to return this item.
- 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. "
- Today isn't Monday and it isn't raining.
- Today isn't Monday or it isn't raining.
- Today isn't Monday or it is raining.
- Today isn't Monday and it is raining.
11. Rewrite each statement using the logical equivalency provided.
- If you were talking, then you missed the instructions. Use \(p \rightarrow q \ \equiv \ \sim p \vee q \).
- It is not true that if Luke faces Vader, then Obi-Wan cannot interfere. Use \(\sim(p \rightarrow q) \ \equiv \ p \ \wedge \sim q \).
- 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. "
- We don't get a pay raise or we are happy.
- We get a pay raise and we are happy.
- We get a pay raise and we aren't happy.
- 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 ."
- You know the password and you can get in.
- You don't know the password or you can get in.
- You don't know the password and you can't get in.
- 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. ”
- Write its converse.
- Write its inverse.
- Write its contrapositive.
15. Consider the conditional statement “ If you are under age 17, then you cannot attend this movie. ”
- Write its converse.
- Write its inverse.
- 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 .”
- If you didn’t use lots of hot sauce, then you didn’t eat that day-old burrito.
- If you don’t eat that day-old burrito, then you won’t use lots of hot sauce.
- If you used lots of hot sauce, then you ate that day-old burrito.
- 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. "
- If I lose my money, then I didn't invest wisely.
- If I don't lose my money, then I invested wisely.
- 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.
- \(\begin{array} {ll} \text{Premise 1:} & p \vee q \\ \text{Premise 2:} & p \\ \text{Conclusion:} & \sim q \end{array}\)
- \(\begin{array} {ll} \text{Premise 1:} & p \rightarrow q \\ \text{Premise 2:} & \sim q \\ \text{Conclusion:} & \sim p \end{array}\)
19. Rewrite each argument symbolically. Then, determine whether the argument is valid or invalid.
- If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.
- Jamie scrubs the toilets or hoses down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie hoses down the garbage cans.
- If I don’t change my oil regularly, then my engine dies. My engine died. Therefore, I didn’t change my oil regularly.