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To determine if this argument is valid, we can analyze the statements using a truth table. The argument presented is:

1. \( p \leftrightarrow \sim q \)
2. \( p \land \sim q \)
3. Therefore, \( p \lor q \)

Let’s break down each component:

- \( p \leftrightarrow \sim q \): This means \( p \) is true if and only if \( \sim q \) (not \( q \)) is true.
- \( p \land \sim q \): This indicates that both \( p \) and \( \sim q \) are true.
- \( p \lor q \): This is the conclusion, which states that either \( p \) or \( q \) must be true.

### Step 1: Build the Truth Table
We will analyze all possible truth values for \( p \) and \( q \), and then compute \( \sim q \), \( p \leftrightarrow \sim q \), \( p \land \sim q \), and \( p \lor q \).

| \( p \) | \( q \) | \( \sim q \) | \( p \leftrightarrow \sim q \) | \( p \land \sim q \) | \( p \lor q \) |
|---------|---------|--------------|---------------------------------|----------------------|----------------|
| T       | T       | F            | F                               | F                    | T              |
| T       | F       | T            | T                               | T                    | T              |
| F       | T       | F            | T                               | F                    | T              |
| F       | F       | T            | F                               | F                    | F              |

### Step 2: Determine Validity
For the argument to be valid, every instance where both premises \( p \leftrightarrow \sim q \) and \( p \land \sim q \) are true should also make the conclusion \( p \lor q \) true.

Looking at the truth table:
- The only row where both premises \( p \leftrightarrow \sim q \) and \( p \land \sim q \) are true is the second row.
- In this row, \( p \lor q \) is also true.

Since there are no cases where the premises are true and the conclusion is false, the argument is **valid**.

Would you like a more detailed explanation of each step? Let me know if you have any questions. 

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Here are five questions to expand on this topic:
1. What does it mean for an argument to be valid in logic?
2. How does the biconditional (\( \leftrightarrow \)) operator work in truth tables?
3. Can we determine the validity of an argument without using a truth table?
4. What is the difference between validity and soundness in logic?
5. How would the argument change if the conclusion were \( \sim(p \land q) \)?

**Tip:** In logic, validity means that if the premises are true, the conclusion must also be true. This does not necessarily mean the premises are actually true; rather, it ensures that the structure of the argument preserves truth.

Using truth tables, determine whether the following argument is valid:
1. p ↔ ~q
2. p ∧ ~q
Therefore, p ∨ q.

Analyze the validity of a logical argument using truth tables. This problem explores propositional logic concepts such as biconditional, conjunction, and disjunction, helping students understand logical validity. Suitable for college-level logic courses.

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