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I can help you complete the truth table from the image.

### Logical symbols explained:
- \( p \): First logical proposition (true or false)
- \( q \): Second logical proposition (true or false)
- \( \sim (p \lor q) \): Negation of the disjunction (OR) of \( p \) and \( q \)
- \( p \land q \): Conjunction (AND) of \( p \) and \( q \)
- \( p \land \sim q \): Conjunction of \( p \) and the negation of \( q \)
- \( \sim p \land \sim q \): Conjunction of the negation of \( p \) and the negation of \( q \)

Let's go step by step through the rows.

### First Row: \( p = T \), \( q = T \)
- \( \sim (p \lor q) \): Since \( p \lor q = T \lor T = T \), \( \sim T = F \)
- \( p \land q \): \( T \land T = T \)
- \( p \land \sim q \): \( \sim q = F \), so \( T \land F = F \)
- \( \sim p \land \sim q \): \( \sim p = F \) and \( \sim q = F \), so \( F \land F = F \)

### Second Row: \( p = T \), \( q = F \)
- \( \sim (p \lor q) \): \( p \lor q = T \lor F = T \), \( \sim T = F \)
- \( p \land q \): \( T \land F = F \)
- \( p \land \sim q \): \( \sim q = T \), so \( T \land T = T \)
- \( \sim p \land \sim q \): \( \sim p = F \) and \( \sim q = T \), so \( F \land T = F \)

### Third Row: \( p = F \), \( q = T \)
- \( \sim (p \lor q) \): \( p \lor q = F \lor T = T \), \( \sim T = F \)
- \( p \land q \): \( F \land T = F \)
- \( p \land \sim q \): \( \sim q = F \), so \( F \land F = F \)
- \( \sim p \land \sim q \): \( \sim p = T \) and \( \sim q = F \), so \( T \land F = F \)

### Fourth Row: \( p = F \), \( q = F \)
- \( \sim (p \lor q) \): \( p \lor q = F \lor F = F \), \( \sim F = T \)
- \( p \land q \): \( F \land F = F \)
- \( p \land \sim q \): \( \sim q = T \), so \( F \land T = F \)
- \( \sim p \land \sim q \): \( \sim p = T \) and \( \sim q = T \), so \( T \land T = T \)

### Filled truth table:

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

Would you like further clarification, or do you have any questions about the steps?

Here are 5 questions to further explore:
1. How is the negation of a disjunction \( \sim (p \lor q) \) related to its components?
2. What patterns do you notice in the conjunction \( p \land q \) when \( p \) and \( q \) change?
3. Can you identify situations where the conjunction \( \sim p \land \sim q \) becomes true?
4. How does the truth table help in proving logical equivalences?
5. How would the table change if you used a different logical operation, such as exclusive OR (XOR)?

**Tip:** When filling in truth tables, always start by calculating the simpler expressions, such as negations, before moving to more complex compound statements.

Fill in the truth table to compare the possible equivalents of logical expressions involving propositions p and q.

Learn how to complete a truth table by comparing logical expressions such as p ∧ q, ¬(p ∨ q), and ¬p ∧ ¬q. This tutorial covers key concepts in logic, including conjunction, disjunction, and negation.

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