https://file.aiquickdraw.com/mathbot/file/1726588950934yqinsbbu.jpg

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.

Determine which of the following are valid arguments: a) p → q, p ∧ q, therefore q. b) I studied. Either I studied or I passed the exam. If I passed the exam, then I did not study. Therefore, I studied.

Learn how to determine the validity of logical arguments using propositional logic. This problem explores Modus Ponens, conjunction elimination, and vacuous truth, providing step-by-step reasoning. Suitable for Grades 10-12.

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