Answered

Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

Select all the correct answers.

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

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

Sagot :

GinnyAnswer
Let's break down the original statement and analyze its inverse.

### Original Statement
The given statement is:
"A number is negative if and only if it is less than 0."
This can be represented in logical terms as:
[tex]\[ p \leftrightarrow q \][/tex]
where,
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.

### Inverse of the Statement
The inverse of a statement is obtained by negating both the hypothesis and the conclusion of the original statement. Thus, the inverse of [tex]\( p \leftrightarrow q \)[/tex] is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
where,
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (which means it is greater than or equal to 0).

### Analyzing the Inverse
We need to determine the truth value of the inverse statement [tex]\( \sim q \rightarrow \sim p \)[/tex].

1. [tex]\( \sim q \)[/tex] (A number is not less than 0) means the number is either 0 or positive.
2. If a number is 0 or positive, [tex]\( \sim p \)[/tex] (A number is not negative) is always true because 0 is non-negative and positive numbers are obviously non-negative.

Therefore, [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically consistent and follows that the inverse statement should be evaluated as false.

So, let's evaluate the provided answers:

- The inverse of the statement is sometimes true and sometimes false: This is incorrect because the inverse is consistently false.
- [tex]\( q \leftrightarrow p \)[/tex]: This represents the original statement or its converse, not its inverse.
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]: This represents the contrapositive of the statement, not its inverse.
- The inverse of the statement is false: This is correct because the inverse is indeed false.
- [tex]\( q \rightarrow p \)[/tex]: This is not the inverse; it is the same as the original implication.
- The inverse of the statement is true: This is incorrect because we have established that the inverse is false.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]: This correctly represents the inverse statement.

### Conclusion:
The correct answers are:
- The inverse of the statement is false.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.