anoroclupita2008
Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Select all the correct answers.

A number is negative if and only if it is less than 0.

Which represents the inverse of this statement? Is the inverse true or false?

A. [tex]\(\sim q \rightarrow \sim p\)[/tex]

B. The inverse of the statement is sometimes true and sometimes false.

C. The inverse of the statement is false.

D. [tex]\(q \leftrightarrow p\)[/tex]

E. [tex]\(q \rightarrow p\)[/tex]

F. [tex]\(\sim p \leftrightarrow \sim q\)[/tex]

G. The inverse of the statement is true.

Sagot :

GinnyAnswer
To solve this problem, let's analyze the given statement, find its inverse, and determine the truth value of that inverse. We have the following statement and its components:

- Original statement: "A number is negative if and only if it is less than 0."
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.

This statement can be written in logical terms as [tex]\( p \leftrightarrow q \)[/tex] (p if and only if q).

### Finding the Inverse of the Statement

The inverse of a statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
However, here we are dealing with [tex]\( p \leftrightarrow q \)[/tex], which is a biconditional statement.

The inverse of [tex]\( p \leftrightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., a number is 0 or positive).
- [tex]\( \sim p \)[/tex]: A number is not negative (i.e., a number is 0 or positive).

Therefore, the inverse statement [tex]\( \sim q \rightarrow \sim p \)[/tex] reads: "If a number is not less than 0, then it is not negative."

### Evaluating the Truth Value of the Inverse

We need to determine if [tex]\( \sim q \rightarrow \sim p \)[/tex] is true or false.

Given:
- If [tex]\( \sim q \)[/tex] means the number is 0 or positive.
- If [tex]\( \sim p \)[/tex] means the number is 0 or positive.

Then the statement "If a number is not less than 0, then it is not negative" holds true in all cases:
- If a number is 0 or positive (not less than 0), it is indeed not negative.

Hence, [tex]\( \sim q \rightarrow \sim p \)[/tex] is true.

### Conclusion

Based on our analysis:

- The inverse of the statement is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- The inverse statement is true.

Thus, the correct options to select are:
1. [tex]\( \sim q \rightarrow \sim p \)[/tex]
7. The inverse of the statement is true.

### Summary

The correct answers are:
[tex]\( 1. \sim q \rightarrow \sim p \)[/tex]
[tex]\( 7. The inverse of the statement is true \)[/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.