To solve this question, let's go through the steps to understand the concepts and the reasoning involved.
### Step 1: Understanding the Original Statement
The original statement is:
- If a figure is a rectangle, it is a parallelogram.
This can be written in conditional form as:
[tex]\[ p \rightarrow q \][/tex]
where
- [tex]\( p \)[/tex] represents "A figure is a rectangle."
- [tex]\( q \)[/tex] represents "A figure is a parallelogram."
### Step 2: Determining the Inverse of the Statement
The inverse of a conditional statement [tex]\( p \rightarrow q \)[/tex] is formed by negating both the hypothesis and the conclusion, which gives us:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
where
- [tex]\( \sim p \)[/tex] represents "A figure is not a rectangle."
- [tex]\( \sim q \)[/tex] represents "A figure is not a parallelogram."
So the inverse statement reads:
If a figure is not a rectangle, it is not a parallelogram.
This can be represented symbolically as:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
### Step 3: Assessing the Truth Value of the Inverse
To determine if the inverse is true or false, consider the definitions:
- A rectangle is a specific type of parallelogram with right angles.
- However, there are many parallelograms that are not rectangles (e.g., rhombuses without right angles).
Because there are figures (parallelograms) that are not rectangles but still parallelograms, the inverse statement "If a figure is not a rectangle, it is not a parallelogram" is false.
### Summary of Answers
Based on our understanding:
1. The inverse of the statement is false. (This is true.)
2. The inverse of the statement is true. (This is false since we established the inverse is false.)
3. [tex]\( p \leftarrow q \)[/tex] (This is not the correct representation of the inverse.)
4. [tex]\( \sim q \leftrightarrow \sim p \)[/tex] (This is not the correct representation of the inverse.)
5. The inverse of the statement is sometimes true and sometimes false. (This is false because we determined the inverse is definitively false.)
6. [tex]\( q \rightarrow p \)[/tex] (This is not the correct representation of the inverse.)
7. [tex]\( \sim p \rightarrow \sim q \)[/tex] (This is the correct symbolic representation of the inverse.)
### Correct Answers:
- The inverse of the statement is false.
- [tex]\(\sim p \rightarrow \sim q\)[/tex]
