how to simplificate ((not(p)or (r)) and ((p) or (q)))?

To simplify the logical expression \(((\neg p \lor r) \land (p \lor q))\), we can apply the **distributive laws of logic** and other simplification rules such as the **identity, dominance, absorption, and negation laws**. Here’s the step-by-step simplification:

### Expression:
\[
(\neg p \lor r) \land (p \lor q)
\]

---

### Step 1: Expand using the distributive property
We distribute each term in \((\neg p \lor r)\) across \((p \lor q)\):
\[
[(\neg p \lor r) \land (p \lor q)] = [(\neg p \land (p \lor q)) \lor (r \land (p \lor q))]
\]

This gives us two separate components:
1. \((\neg p \land (p \lor q))\)
2. \((r \land (p \lor q))\)

---

### Step 2: Simplify the first term \((\neg p \land (p \lor q))\)
Distribute \(\neg p\) across \((p \lor q)\):
\[
(\neg p \land (p \lor q)) = (\neg p \land p) \lor (\neg p \land q)
\]

- \((\neg p \land p) = \text{False}\) (Contradiction Law)
- \((\neg p \land q) = (\neg p \land q)\)

Thus:
\[
(\neg p \land (p \lor q)) = (\neg p \land q)
\]

---

### Step 3: Simplify the second term \((r \land (p \lor q))\)
Distribute \(r\) across \((p \lor q)\):
\[
(r \land (p \lor q)) = (r \land p) \lor (r \land q)
\]

---

### Step 4: Combine the results
Now we combine the simplified terms:
\[
(\neg p \lor r) \land (p \lor q) = (\neg p \land q) \lor (r \land p) \lor (r \land q)
\]

---

### Step 5: Factor and simplify further if possible
The expression can be left in its simplified form:
\[
(\neg p \land q) \lor (r \land p) \lor (r \land q)
\]

Alternatively, you could write it as:
\[
(r \land (p \lor q)) \lor (\neg p \land q)
\]

This is the simplest form.

---

### Final Answer:
\[
(\neg p \land q) \lor (r \land p) \lor (r \land q)
\]

---

Do you want me to break down any specific step further? 😊

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### Related Questions:
1. What are the primary laws of logic used in this simplification?
2. Can you explain why \((\neg p \land p)\) simplifies to \(\text{False}\)?
3. How does distributive property in logic compare to distributive property in algebra?
4. How would you express this simplified formula as a truth table?
5. Can this expression be further reduced using Karnaugh maps?

### Tip:
When simplifying logical expressions, always identify terms that result in contradictions (\(p \land \neg p\)) or tautologies (\(p \lor \neg p\)) to quickly eliminate unnecessary components.

how to simplificate ((not(p) or (r)) and ((p) or (q)))?

Discover how to simplify the logical expression ((¬p ∨ r) ∧ (p ∨ q)) using Boolean algebra. This guide covers distributive, contradiction, and absorption laws with clear step-by-step explanations. Suitable for high school students in Grades 10-12.

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