The problem concerns logical reasoning and understanding of the inverse of a conditional statement. Let me break it down:

Given Statement:

If a figure is a rectangle (p), then it is a parallelogram (q).

This can be written as: $p \to q$

Inverse of a Statement:

The inverse of a conditional statement $p \to q$ is: $\neg p \to \neg q$ This means: If a figure is not a rectangle, then it is not a parallelogram.

Now, analyzing the choices:

1. q → p: This is the converse, not the inverse.
2. The inverse of the statement is sometimes true and sometimes false.: This depends on whether the inverse matches reality.
3. The inverse of the statement is true.: This needs verification with the given conditions.
4. The inverse of the statement is false.: Again, this needs verification.
5. ∼p → ∼q: This is the correct symbolic representation of the inverse.

We also assess if the inverse is true or false:

• A rectangle is always a parallelogram, but the reverse (whether the inverse is valid) depends on whether "not being a rectangle means not being a parallelogram," which is generally false (e.g., rhombuses and general parallelograms exist).

Answer:

• The correct symbolic representation of the inverse is $\sim p \to \sim q$.
• The inverse of the statement is false because there exist non-rectangular parallelograms.

Let me know if you want further explanation or clarification!

Expanding Questions:

1. What is the difference between the converse and inverse of a statement?
2. How would the contrapositive of the given statement be written?
3. In logical reasoning, when is the inverse of a statement guaranteed to be true?
4. Can you give real-world examples where an inverse is true?
5. How does understanding logical inverses help in mathematical proofs?

Tip: Always verify logical statements with counterexamples to check for truth values.