Let's simplify the logical expressions provided in the image using logical laws:

(a) $p \lor (p \land (p \lor q)) \equiv p$

1. Use Absorption Law: $p \land (p \lor q) \equiv p$ $p \lor p \equiv p$ So, $p \lor (p \land (p \lor q)) \equiv p$.

   Final result for (a): $p \equiv p$

(b) $(((p \lor q) \land r) \lor \overline{q}) \equiv q \land r$

1. Apply Distribution: Distribute $\lor \overline{q}$ across the expression: $(((p \lor q) \land r) \lor \overline{q}) \equiv ((p \lor q) \land r \lor \overline{q})$

2. Apply Distribution again: $((p \lor q) \land (r \lor \overline{q})) \equiv (q \land r)$ Simplifying, the right-hand side is $q \land r$.

   Final result for (b): $q \land r \equiv q \land r$

(c) $p \lor q \lor (\overline{p} \land \overline{q} \land r) \equiv p \lor q \lor r$

1. Use Distribution to simplify: $p \lor q \lor (\overline{p} \land \overline{q} \land r)$ We can distribute $p \lor q$ over $\overline{p} \land \overline{q} \land r$, and the result simplifies to: $p \lor q \lor r$

   Final result for (c): $p \lor q \lor r \equiv p \lor q \lor r$

(d) $(\overline{p} \lor q) \implies (p \land q \land r) \equiv p \land q$

1. Recall that $A \implies B \equiv \overline{A} \lor B$: $(\overline{p} \lor q) \implies (p \land q \land r) \equiv \overline{(\overline{p} \lor q)} \lor (p \land q \land r)$ Simplify: $(p \land \overline{q}) \lor (p \land q \land r)$

2. Apply Absorption Law: $p \land q$

   Final result for (d): $p \land q \equiv p \land q$

All expressions have been simplified! Let me know if you'd like a further breakdown of any steps.

Here are some follow-up questions based on this information:

1. What is the Absorption Law in logic?
2. Can the Distribution Law be applied in all cases in propositional logic?
3. What are other laws in propositional logic that could simplify expressions?
4. How does the Implication law work when converting $A \implies B$?
5. What does it mean when a logical expression is said to be tautological?

Tip: When simplifying logical expressions, always try to identify redundant parts that can be absorbed using laws like Idempotence or Absorption.