Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's carefully work through the problem.
Given statement: A number is negative if and only if it is less than 0.
- p: A number is negative.
- q: A number is less than 0.
To represent the inverse of this statement:
1. The inverse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., a number is greater than or equal to 0).
Thus, the inverse statement is:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
Now let's evaluate the truth of the inverse statement.
- When [tex]\( p \)[/tex] is false, [tex]\( \sim p \)[/tex] is true (the number is not negative).
- When [tex]\( q \)[/tex] is false, [tex]\( \sim q \)[/tex] is true (the number is not less than 0).
### Truth Evaluation for the Inverse Statement:
- [tex]\( \sim p \rightarrow \sim q \)[/tex] holds true when both [tex]\( \sim p \)[/tex] and [tex]\( \sim q \)[/tex] are true (i.e., the number is not negative and is greater than or equal to 0).
- [tex]\( \sim p \rightarrow \sim q \)[/tex] is false when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
To summarize:
- The inverse of the statement [tex]\( \sim p \rightarrow \sim q \)[/tex] is true if both the number is not negative and the number is greater than or equal to 0 hold together.
- The inverse is false if any one of these conditions fails to hold when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
Let's review the possible answers:
1. [tex]\( \sim p \leftrightarrow \sim q \)[/tex] - This is not the inverse of the original statement.
2. [tex]\( q \leftrightarrow p \)[/tex] - This is not the inverse of the original statement.
3. The inverse of the statement is sometimes true and sometimes false. - This is not correct because we determined that the inverse statement [tex]\( \sim p \rightarrow \sim q \)[/tex] can be evaluated as generally true under the given definitions.
4. The inverse of the statement is false. - This is incorrect; the logical interpretation shows it can be true.
5. [tex]\( \sim q \rightarrow \sim p \)[/tex] - This is incorrect.
6. The inverse of the statement is true. - This is correct.
7. [tex]\( q \rightarrow p \)[/tex] - This is not the inverse of the original statement.
Based on these evaluations, the correct answers are:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
[tex]\[ \text{The inverse of the statement is true.} \][/tex]
Thus, the correct answers are:[tex]\[ \textbf{The inverse of the statement is true.} \][/tex]
[tex]\[ \sim p \rightarrow \sim q \][/tex]
Given statement: A number is negative if and only if it is less than 0.
- p: A number is negative.
- q: A number is less than 0.
To represent the inverse of this statement:
1. The inverse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., a number is greater than or equal to 0).
Thus, the inverse statement is:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
Now let's evaluate the truth of the inverse statement.
- When [tex]\( p \)[/tex] is false, [tex]\( \sim p \)[/tex] is true (the number is not negative).
- When [tex]\( q \)[/tex] is false, [tex]\( \sim q \)[/tex] is true (the number is not less than 0).
### Truth Evaluation for the Inverse Statement:
- [tex]\( \sim p \rightarrow \sim q \)[/tex] holds true when both [tex]\( \sim p \)[/tex] and [tex]\( \sim q \)[/tex] are true (i.e., the number is not negative and is greater than or equal to 0).
- [tex]\( \sim p \rightarrow \sim q \)[/tex] is false when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
To summarize:
- The inverse of the statement [tex]\( \sim p \rightarrow \sim q \)[/tex] is true if both the number is not negative and the number is greater than or equal to 0 hold together.
- The inverse is false if any one of these conditions fails to hold when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
Let's review the possible answers:
1. [tex]\( \sim p \leftrightarrow \sim q \)[/tex] - This is not the inverse of the original statement.
2. [tex]\( q \leftrightarrow p \)[/tex] - This is not the inverse of the original statement.
3. The inverse of the statement is sometimes true and sometimes false. - This is not correct because we determined that the inverse statement [tex]\( \sim p \rightarrow \sim q \)[/tex] can be evaluated as generally true under the given definitions.
4. The inverse of the statement is false. - This is incorrect; the logical interpretation shows it can be true.
5. [tex]\( \sim q \rightarrow \sim p \)[/tex] - This is incorrect.
6. The inverse of the statement is true. - This is correct.
7. [tex]\( q \rightarrow p \)[/tex] - This is not the inverse of the original statement.
Based on these evaluations, the correct answers are:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
[tex]\[ \text{The inverse of the statement is true.} \][/tex]
Thus, the correct answers are:[tex]\[ \textbf{The inverse of the statement is true.} \][/tex]
[tex]\[ \sim p \rightarrow \sim q \][/tex]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
