Let's analyze the two arguments provided in the image to determine if they are valid.

Argument (a)

The structure of this argument is:

1. $p \rightarrow q$ (If $p$, then $q$)
2. $p \land q$ (Both $p$ and $q$ are true)
3. Therefore, $q$

This argument is valid because the premises directly support the conclusion.

• $p \rightarrow q$ means if $p$ is true, $q$ must be true.
• $p \land q$ states both $p$ and $q$ are true, so $q$ is confirmed to be true.

This is an example of modus ponens and conjunction elimination, both valid logical inferences.

Argument (b)

The structure of this argument is more verbal:

1. "I studied."
2. "Either I studied or I passed the exam."
3. "If I passed the exam, then I did not study."

Conclusion: "I studied."

This argument is a bit tricky. We can formalize it as follows:

• Let $S$ represent "I studied" and $P$ represent "I passed the exam."
• Premise 1: $S$ (I studied).
• Premise 2: $S \lor P$ (Either I studied or I passed the exam).
• Premise 3: $P \rightarrow \neg S$ (If I passed the exam, then I did not study).

In this case, the argument is also valid because:

• Premise 1 already asserts $S$ (I studied), which satisfies the conclusion directly.
• The third premise $P \rightarrow \neg S$ doesn't contradict this, as there's no indication that $P$ is true; in fact, $P$ must be false because $S$ is true (as stated in premise 1). Therefore, $P \rightarrow \neg S$ is vacuously true.

Conclusion:

• Both arguments (a) and (b) are valid. Thus, the correct answer is A) Both a and b.

Would you like further details on any step of this analysis?

Here are 5 related questions to further explore this topic:

1. What is the difference between a valid argument and a sound argument?
2. Can you explain how modus ponens works in more detail?
3. How does the truth table for $p \rightarrow q$ look?
4. What are some common logical fallacies in reasoning?
5. How can a vacuous truth make an argument valid?

Tip: When analyzing logical arguments, always look for the structure and relationships between premises to determine their validity.