https://file.aiquickdraw.com/mathbot/file/1726759131441os7t58gx.jpg

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**.

Simplify the given logical expressions using logical laws.

Learn how to simplify complex logical expressions using various laws of propositional logic, including Absorption, Distribution, and Implication laws. This detailed guide is designed for college-level students who are studying formal logic and discrete mathematics.

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