Use the set of identities to show that ¬(pV (¬p∧q)) ≡(¬p ∧¬𝒒)

To show that \( \neg(p \vee (\neg p \land q)) \equiv (\neg p \land \neg q) \), we’ll use logical identities step-by-step to prove the equivalence. Here's the breakdown:

### Step 1: Apply De Morgan’s law to the outer negation
Starting with:
\[
\neg(p \vee (\neg p \land q))
\]
Using De Morgan’s law, which states \( \neg(A \vee B) \equiv \neg A \land \neg B \), we get:
\[
\neg(p) \land \neg(\neg p \land q)
\]

### Step 2: Apply De Morgan’s law to the inner negation
Now we simplify \( \neg(\neg p \land q) \) using De Morgan’s law again, which gives:
\[
\neg(\neg p \land q) \equiv \neg(\neg p) \vee \neg(q)
\]
Since \( \neg(\neg p) \equiv p \), this simplifies to:
\[
p \vee \neg q
\]
Thus, we now have:
\[
\neg(p) \land (p \vee \neg q)
\]

### Step 3: Distribute \( \neg p \) over \( (p \vee \neg q) \) using the distributive law
Using the distributive law, \( A \land (B \vee C) \equiv (A \land B) \vee (A \land C) \), we get:
\[
(\neg p \land p) \vee (\neg p \land \neg q)
\]

### Step 4: Simplify using the contradiction law
Notice that \( \neg p \land p \) is a contradiction (it is always false), so:
\[
(\neg p \land p) \vee (\neg p \land \neg q) \equiv \text{false} \vee (\neg p \land \neg q)
\]
Since \( \text{false} \vee X \equiv X \), we get:
\[
\neg p \land \neg q
\]

### Conclusion
We have shown that:
\[
\neg(p \vee (\neg p \land q)) \equiv \neg p \land \neg q
\]
This completes the proof.

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Would you like more details or have any questions?

Here are some related questions that expand on this topic:
1. What are the main laws of propositional logic used in simplifications?
2. How can De Morgan's laws help in simplifying complex logical statements?
3. How does the distributive law apply in Boolean algebra?
4. How can contradiction laws help in logical simplifications?
5. Why is understanding logical equivalences important in proofs and problem-solving?

**Tip:** Mastering De Morgan's laws is essential for simplifying negated statements in logic.

Use the set of identities to show that ¬(p ∨ (¬p ∧ q)) ≡ (¬p ∧ ¬q)

Follow this step-by-step solution to show that ¬(p ∨ (¬p ∧ q)) ≡ (¬p ∧ ¬q) by applying propositional logic identities, including De Morgan's laws, the distributive law, and the contradiction law. Ideal for college students studying Boolean Algebra and logical equivalences.

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