To simplify the logical expression $\neg p \lor r \land (p \lor q)$, we will use standard logical equivalences:

Expression:

$\neg p \lor (r \land (p \lor q))$

Step 1: Apply Distributive Law over $\lor$ and $\land$:

Using the rule $A \lor (B \land C) = (A \lor B) \land (A \lor C)$, distribute $\neg p$ over $r \land (p \lor q)$: $(\neg p \lor r) \land (\neg p \lor (p \lor q))$

Step 2: Simplify $(\neg p \lor (p \lor q))$:

By associativity of $\lor$, $(\neg p \lor (p \lor q)) = (\neg p \lor p \lor q)$. Then apply the tautology $p \lor \neg p = \text{True}$, so: $\neg p \lor (p \lor q) = \text{True} \lor q = \text{True}.$

Thus, the expression becomes: $(\neg p \lor r) \land \text{True}.$

Step 3: Eliminate the conjunction with True:

Anything conjoined with $\text{True}$ is itself, so: $\neg p \lor r$

Final Simplified Expression:

$\neg p \lor r$

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Would you like me to expand on logical equivalences, or do you have further questions?

Here are some related questions to deepen your understanding:

1. What are the fundamental logical equivalences used in simplifying logical expressions?
2. How does the distributive law work in Boolean algebra?
3. Can you simplify expressions with multiple $\land$ and $\lor$ operations?
4. What does it mean when an expression simplifies to $\text{True}$ or $\text{False}$?
5. How can you construct truth tables to verify simplified expressions?

Tip: Always try to identify tautologies and contradictions early to reduce the complexity of logical expressions.