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Select all the correct answers.

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.

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

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

Sagot :

GinnyAnsweridn
To find the correct answers, we need to understand the original statement and its inverse. Let's break this down step-by-step:

1. Original Statement:
- "A number is negative if and only if it is less than 0."
- This means [tex]\( p \leftrightarrow q \)[/tex].
- Where [tex]\( p \)[/tex]: A number is negative.
- And [tex]\( q \)[/tex]: A number is less than 0.

2. Inverse of the Statement:
- The inverse of a statement [tex]\( p \leftrightarrow q \)[/tex] is [tex]\( \sim q \leftrightarrow \sim p \)[/tex].
- Here, [tex]\( \sim p \)[/tex]: A number is not negative.
- And [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., it is 0 or positive).

3. Evaluating the Inverse:
- Let's break down [tex]\( \sim q \leftrightarrow \sim p \)[/tex]:

[tex]\( \sim q \)[/tex]: A number is not less than 0.
 [tex]\( \sim p \)[/tex]: A number is not negative.

- So, the inverse statement [tex]\( \sim q \leftrightarrow \sim p \)[/tex] translates to:
"A number is not less than 0 if and only if it is not negative."

- This statement is true, because:
If a number is not less than 0 (i.e., it’s 0 or positive), then it is not negative.
 If a number is not negative, then it is not less than 0 (i.e., it’s 0 or positive).

So, the inverse statement [tex]\( \sim q \leftrightarrow \sim p \)[/tex] is true.

4. Summary of Options:
- The inverse of the statement is true. (Correct)
- [tex]\( q \leftrightarrow p \)[/tex]. (Incorrect) – This is the original statement.
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]. (Correct) – This is the correct representation of the inverse.
- The inverse of the statement is sometimes true and sometimes false. (Incorrect) – The inverse is always true, not sometimes.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]. (Incorrect) – This represents only part of the inverse statement; it is not the full biconditional inverse.
- [tex]\( q \rightarrow p \)[/tex]. (Incorrect) – This is part of the original statement, not the inverse.
- The inverse of the statement is false. (Incorrect) – The inverse is true, not false.

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