Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Let \( p = \) "2 is a factor of \( n \)", \( q = \) "3 is a factor of \( n \)", and \( r = \) "6 is a factor of \( n \)."
**(a)** Let Statement A be "If 2 is a factor of \( n \) and 3 is a factor of \( n \), then 6 is a factor of \( n \)." Which of the following expresses Statement A in symbolic form?
- \( p \lor q \)
- \( p \land q \rightarrow r \)
- \( p \lor q \rightarrow r \)
- \( \neg p \land q \rightarrow r \)
- \( p \land \neg q \)
**(b)** Let Statement B be "2 is not a factor of \( n \), or 3 is not a factor of \( n \), or 6 is a factor of \( n \)." Which of the following expresses Statement B in symbolic form?
- \( \neg p \land \neg q \land r \)
- \( p \lor \neg r \)
- \( \neg p \lor r \)
- \( \neg p \land \neg q \rightarrow r \)
- \( \neg p \lor \neg q \lor r \)
**(c)** Fill in the values in the truth table for Statements A and B.
\[
\begin{array}{ccccccc}
p & q & r & \neg q & p \land q & p \land q \rightarrow r & \neg p \lor \neg q \lor r \\
\hline
T & T & T & & & & \\
T & T & F & & & & \\
T & F & T & & & & \\
T & F & F & & & & \\
F & T & T & & & & \\
F & T & F & & & & \\
F & F & T & & & & \\
F & F & F & & & & \\
\end{array}
\]
**Question:** Are Statements A and B logically equivalent? Fill in the blanks in the sentences below to justify your answer.
The truth table shows that \( p \land q \rightarrow r \) and \( \neg p \lor \neg q \lor r \) \_\_\_\_\_ have the same truth values](https://content.bartleby.com/qna-images/question/7cff8c4a-bb33-4b54-bd2d-8ccd43ef3f82/73519185-20ac-42b4-8651-13ae430baef2/th584xx_processed.png)
Transcribed Image Text:Let \( p = \) "2 is a factor of \( n \)", \( q = \) "3 is a factor of \( n \)", and \( r = \) "6 is a factor of \( n \)."
**(a)** Let Statement A be "If 2 is a factor of \( n \) and 3 is a factor of \( n \), then 6 is a factor of \( n \)." Which of the following expresses Statement A in symbolic form?
- \( p \lor q \)
- \( p \land q \rightarrow r \)
- \( p \lor q \rightarrow r \)
- \( \neg p \land q \rightarrow r \)
- \( p \land \neg q \)
**(b)** Let Statement B be "2 is not a factor of \( n \), or 3 is not a factor of \( n \), or 6 is a factor of \( n \)." Which of the following expresses Statement B in symbolic form?
- \( \neg p \land \neg q \land r \)
- \( p \lor \neg r \)
- \( \neg p \lor r \)
- \( \neg p \land \neg q \rightarrow r \)
- \( \neg p \lor \neg q \lor r \)
**(c)** Fill in the values in the truth table for Statements A and B.
\[
\begin{array}{ccccccc}
p & q & r & \neg q & p \land q & p \land q \rightarrow r & \neg p \lor \neg q \lor r \\
\hline
T & T & T & & & & \\
T & T & F & & & & \\
T & F & T & & & & \\
T & F & F & & & & \\
F & T & T & & & & \\
F & T & F & & & & \\
F & F & T & & & & \\
F & F & F & & & & \\
\end{array}
\]
**Question:** Are Statements A and B logically equivalent? Fill in the blanks in the sentences below to justify your answer.
The truth table shows that \( p \land q \rightarrow r \) and \( \neg p \lor \neg q \lor r \) \_\_\_\_\_ have the same truth values
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