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The question asks whether the inverse and the contrapositive of a conditional statement can both have false truth values, and you're required to explain this.

Let's break down the question based on the image.

### Inverse and Contrapositive:
- **Inverse** of the conditional "if \( p \), then \( q \)" (written \( p \rightarrow q \)) is written as "if not \( p \), then not \( q \)" (symbolically, \( \neg p \rightarrow \neg q \)).
- **Contrapositive** of the conditional "if \( p \), then \( q \)" is written as "if not \( q \), then not \( p \)" (symbolically, \( \neg q \rightarrow \neg p \)).

### Truth Values of the Inverse and Contrapositive:
1. The **inverse** \( \neg p \rightarrow \neg q \) is false when \( \neg p \) is true (i.e., \( p \) is false) and \( \neg q \) is false (i.e., \( q \) is true).
2. The **contrapositive** \( \neg q \rightarrow \neg p \) is false when \( \neg q \) is true (i.e., \( q \) is false) and \( \neg p \) is false (i.e., \( p \) is true).

### Filling in the Blanks:
- The inverse \( \neg p \rightarrow \neg q \) is false when \( p \) is false and \( q \) is true.
- The contrapositive \( \neg q \rightarrow \neg p \) is false when \( p \) is true and \( q \) is false.

### Conclusion:
- It is **not possible** for both the inverse and the contrapositive to be false simultaneously because for the inverse to be false, \( p \) must be false and \( q \) true, whereas for the contrapositive to be false, \( p \) must be true and \( q \) false. These conditions cannot occur at the same time.

Would you like further details or have any questions? Here are some related questions:

1. What is the truth table for the inverse and contrapositive?
2. How does the truth table for a conditional statement differ from that of its inverse?
3. Can the contrapositive of a conditional statement ever have the same truth value as the original?
4. Under what conditions are the inverse and the converse both true?
5. How do you prove that a contrapositive is logically equivalent to the original statement?

**Tip:** The contrapositive of a conditional statement always has the same truth value as the original conditional.

Can an inverse and the contrapositive both have false truth values? Explain.

Explore whether the inverse and contrapositive of a conditional statement can both have false truth values. This problem covers logic, truth tables, and conditional statements, providing an explanation of inverse and contrapositive relationships. Suitable for students in Grades 9-12.

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