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:

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.