Detailed Video Solution: https://youtu.be/WjixNxzbkAQ
A. For any propositions $\text{P}$ and $\text{Q},$ the following is always true: $(\text{P} \rightarrow \text{Q}) \vee (\text{Q} \rightarrow \text{P}).$
$\textbf{Proof}:$
Here's one way to see this. If $\text{Q}$ is true, then $\text{P} \rightarrow \text{Q}$ is true because anything implies a true statement. If $\text{Q}$ is false, then $\text{Q} \rightarrow \text{P}$ is true because false implies anything. (If this is confusing, you should review the truth table for $\rightarrow )$
B. For any propositions $\text{P}, \text{Q},$ and $\text{R},$ the following statement is always true: $(\text{P} \rightarrow \text{Q}) \vee (\text{Q} \rightarrow \text{R}).$
$\textbf{Proof}:$
This is basically the same argument as before. If $\text{Q}$ is true, then $\text{P} \rightarrow \text{Q}$
is true because anything implies a true statement. If $\text{Q}$ is false, then $\text{Q} \rightarrow \text{R}$ is true because false implies anything.
Of all the connectives we've seen, the $\rightarrow$ connective is probably the trickiest. We asked this question to force you to disentangle notions of correlation or causality from the behavior of the $\rightarrow$ connective.
