To answer whether $p - r$ is false given that $p - q$ is true and $q - r$ is false, we need to examine what the notation means.

Assuming that the expressions represent logical implications:

• $p - q$ means "if $p$, then $q$" (i.e., $p \implies q$).
• $q - r$ means "if $q$, then $r$" (i.e., $q \implies r$).

Given Information:

• $p - q$ (or $p \implies q$) is true.
• $q - r$ (or $q \implies r$) is false.

For $q - r$ to be false, the only scenario where an implication is false is when the premise is true but the conclusion is false. So, we deduce:

• $q$ is true.
• $r$ is false.

Now, we use this information to check $p - r$ (i.e., $p \implies r$).

Since $r$ is false, for $p - r$ to be true, $p$ must also be false (because the only situation where an implication is false is when the premise is true and the conclusion is false).

Conclusion:

• If $p$ were true, then $p \implies r$ would be false because $r$ is false.
• Therefore, $p - r$ is false in this case.

Summary:

Yes, $p - r$ is false given that $p - q$ is true and $q - r$ is false.

Would you like further details on logical implications or clarifications?

Here are 5 relative questions for further exploration:

1. What is the truth table for logical implications?
2. Can two false premises in a chain of implications lead to a true conclusion?
3. How does contrapositive reasoning work in logical implications?
4. What are common examples where implications are used in real-world scenarios?
5. Can we deduce anything else about $p$ based on the given conditions?

Tip: Remember that a logical implication is false only when the premise is true and the conclusion is false.