Let's break down each part of the problem methodically.
1. Understanding the Statements:
- The original statement is: "If a figure is a rectangle, it is a parallelogram."
- In logical form: [tex]\(p \to q\)[/tex]
- Where [tex]\(p\)[/tex] stands for "A figure is a rectangle."
- And [tex]\(q\)[/tex] stands for "A figure is a parallelogram."
2. Finding the Inverse:
- The inverse of a statement [tex]\(p \to q\)[/tex] is [tex]\(\sim p \to \sim q\)[/tex].
- Where [tex]\(\sim p\)[/tex] means "A figure is not a rectangle."
- And [tex]\(\sim q\)[/tex] means "A figure is not a parallelogram."
3. Checking the Inverse:
- The inverse of the original statement is: "If a figure is not a rectangle, it is not a parallelogram."
- In logical form: [tex]\(\sim p \to \sim q\)[/tex].
4. Determining the Truth Value of the Inverse:
- The original statement says: "If a figure is a rectangle, it is a parallelogram."
- This is true because all rectangles are parallelograms.
- The inverse statement says: "If a figure is not a rectangle, it is not a parallelogram."
- This is false because there are figures that are not rectangles but are still parallelograms (for example, rhombuses and generic parallelograms that are not rectangles).
Given this analysis, let's match the options provided:
- [tex]\(\sim q \leftrightarrow \sim p\)[/tex]: This represents the contrapositive of the inverse statement, which is not what we are looking for.
- [tex]\(q \to p\)[/tex]: This is the converse of the original statement, which is also not what we are looking for.
- The inverse of the statement is true: This option is incorrect.
- The inverse of the statement is sometimes true and sometimes false: This option is not appropriate because we are strictly determining if the inverse is true or false.
- [tex]\(p \leftarrow q\)[/tex]: This is simply another notation for [tex]\(q \to p\)[/tex], the converse, which is not relevant here.
- The inverse of the statement is false: This is correct. We determined that the inverse statement is false.
- [tex]\(\sim p \to \sim q\)[/tex]: This is indeed the correct form of the inverse statement.
So, the options that are correct are:
- The inverse of the statement is false.
- [tex]\(\sim p \rightarrow \sim q\)[/tex].
