To determine the truth value of the statement $\sim \left( p \lor (r \lor \sim p) \right)$, let's break down each part of the expression step-by-step, given that:

• $p$ is false.
• $r$ is true.

The notation $\sim$ represents negation, and $\lor$ represents the logical "or" operation.

Step-by-Step Evaluation

1. Evaluate $\sim p$:

   • Since $p$ is false, $\sim p$ is true.

2. Evaluate $r \lor \sim p$:

   • $r$ is true and $\sim p$ is true.
   • The expression $r \lor \sim p$ is true (since at least one operand in an "or" operation is true).

3. Evaluate $p \lor (r \lor \sim p)$:

   • $p$ is false, and we already found that $r \lor \sim p$ is true.
   • Therefore, $p \lor (r \lor \sim p)$ is true.

4. Negate the entire expression:

   • We now have $\sim \left( p \lor (r \lor \sim p) \right)$, which is the negation of a true statement.
   • Negating true gives us false.

Conclusion

The truth value of the expression $\sim \left( p \lor (r \lor \sim p) \right)$ is false.

Would you like further details on any specific part? Here are some additional questions to explore the topic:

1. What are the properties of logical negation and disjunction?
2. How does the truth table for "or" operations look for different truth values?
3. How does the order of operations affect logical statements?
4. Can you determine the truth value of a compound statement with different truth values for each variable?
5. How does De Morgan's law apply to compound logical statements?

Tip: When working with complex logical statements, evaluate from the innermost parentheses outward for clarity.