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To determine whether the argument is valid, let’s analyze it using a truth table.

The argument given is:
1. \( p \rightarrow q \)
2. \( p \land q \)
3. Therefore, \( p \)

This is an example of an argument with premises \( p \rightarrow q \) and \( p \land q \), aiming to conclude \( p \). To validate this, we will construct a truth table that includes all possible truth values for \( p \) and \( q \), and evaluate the statements step-by-step.

### Step 1: Set up the Truth Table

| \( p \) | \( q \) | \( p \rightarrow q \) | \( p \land q \) | Conclusion: \( p \) |
|---------|---------|-----------------------|-----------------|----------------------|
| T       | T       | T                     | T               | T                    |
| T       | F       | F                     | F               | T                    |
| F       | T       | T                     | F               | F                    |
| F       | F       | T                     | F               | F                    |

### Step 2: Evaluate Validity

To determine if the argument is valid, we check if every time the premises (\( p \rightarrow q \) and \( p \land q \)) are true, the conclusion \( p \) must also be true.

- Row 1: When \( p \) and \( q \) are both true, both premises \( p \rightarrow q \) and \( p \land q \) are true, and the conclusion \( p \) is also true.
- Row 2: When \( p \) is true and \( q \) is false, \( p \rightarrow q \) is false, making the argument invalid in this case. 

Since we found a case (row 2) where the premises do not guarantee the conclusion, the argument is **not valid**.

Would you like further details, or have any questions?

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Here are 5 related questions to consider:
1. What is the meaning of logical implication \( p \rightarrow q \) in truth tables?
2. How can we formally define the validity of an argument using truth tables?
3. What are the conditions for an argument to be considered valid?
4. Why does the row with \( p = T \) and \( q = F \) invalidate the argument?
5. How can we apply this truth table approach to more complex arguments?

**Tip:** To analyze an argument’s validity with a truth table, ensure all rows where the premises are true also make the conclusion true.

Using truth tables, determine whether the following argument is valid:
p → q
p ∧ q
∴ p

Analyze the validity of the argument 'p → q, p ∧ q, Therefore p' using a detailed truth table. This problem explores key logical concepts such as implications, conjunctions, and how truth tables are used to assess the validity of arguments. Suitable for high school students in Grades 10-12.

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