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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]
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