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Select all the correct answers.

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.

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

A. The inverse of the statement is false.
B. The inverse of the statement is sometimes true and sometimes false.
C. [tex]q \leftrightarrow p[/tex]
D. The inverse of the statement is true.
E. [tex]q \rightarrow p[/tex]
F. [tex]\sim p \leftrightarrow \sim q[/tex]
G. [tex]\sim q \rightarrow \sim p[/tex]


Sagot :

To solve this problem, we will first identify the statement given and then determine its inverse logically.

### Original Statement:
The statement given is:
"A number is negative if and only if it is less than 0."

Let's denote:
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.

The original statement can be written as:
[tex]\[ p \leftrightarrow q \][/tex]
This denotes a biconditional statement meaning "p if and only if q." In other words, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are equivalent.

### Inverse of the Statement:
The inverse of a statement [tex]\(\rightarrow\)[/tex] is formed by negating both the hypothesis and the conclusion. For a biconditional statement [tex]\( p \leftrightarrow q \)[/tex], the inverse will be:
[tex]\[ \neg p \leftrightarrow \neg q \][/tex]

This reads as:
"A number is not negative if and only if it is not less than 0."

### Determine the Truth Value of the Inverse:
To check the truth value of the inverse, we consider whether the inverse statement holds in all conditions:

- The statement [tex]\( p \leftrightarrow q \)[/tex] means [tex]\( p \)[/tex] is true if and only if [tex]\( q \)[/tex] is true.
- If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true and if [tex]\( p \)[/tex] is false, then [tex]\( q \)[/tex] must be false.

Therefore, the negation of both statements will also represent a true relationship:
- If a number is not negative (i.e., [tex]\(\neg p\)[/tex]), then it is not less than 0 (i.e., [tex]\(\neg q\)[/tex]).

Thus, the inverse statement [tex]\(\neg p \leftrightarrow \neg q\)[/tex] remains true.

### Summary of Correct Answers:
- The inverse of the statement is true.
- [tex]\(\sim p \leftrightarrow \sim q\)[/tex]

Given our analysis, the correct selections are:
- The inverse of the statement is true.
- [tex]\(\sim p \leftrightarrow \sim q\)[/tex]

The other options do not accurately describe the inverse statement or its truth value.
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