without using truth table prove (~pvq)^(p^(P^q))

To prove the logical equivalence of the expression \((\neg p \vee q) \wedge (p \wedge (\neg p \vee q))\), we can simplify the expression using logical identities. Here's the step-by-step proof:

### Step 1: Simplify the expression
Consider the given expression:

\[
(\neg p \vee q) \wedge (p \wedge (\neg p \vee q))
\]

This expression is a conjunction (\(\wedge\)) of two components: \((\neg p \vee q)\) and \((p \wedge (\neg p \vee q))\).

### Step 2: Distribute \((\neg p \vee q)\) over the conjunction
The expression can be expanded by distributing \((\neg p \vee q)\) over the conjunction:

\[
[(\neg p \vee q) \wedge p] \wedge [(\neg p \vee q) \wedge (\neg p \vee q)]
\]

Since \((\neg p \vee q) \wedge (\neg p \vee q)\) simplifies to \((\neg p \vee q)\), the expression becomes:

\[
[(\neg p \vee q) \wedge p] \wedge (\neg p \vee q)
\]

### Step 3: Simplify the conjunction \((\neg p \vee q) \wedge p\)
Now, consider the conjunction \((\neg p \vee q) \wedge p\). This can be rewritten using the distributive law:

\[
[(\neg p \wedge p) \vee (q \wedge p)]
\]

Since \(\neg p \wedge p\) is a contradiction (i.e., always false), it simplifies to:

\[
\text{false} \vee (q \wedge p)
\]

This further simplifies to:

\[
q \wedge p
\]

Thus, the original expression simplifies to:

\[
(q \wedge p) \wedge (\neg p \vee q)
\]

### Step 4: Distribute \(q \wedge p\) over the conjunction
Distribute \(q \wedge p\) over the conjunction:

\[
[q \wedge p \wedge \neg p] \vee [q \wedge p \wedge q]
\]

### Step 5: Simplify each term
1. The term \(q \wedge p \wedge \neg p\) is a contradiction, as \(p\) and \(\neg p\) cannot both be true, so this term simplifies to false.
   
2. The term \(q \wedge p \wedge q\) simplifies to \(q \wedge p\), because \(q \wedge q\) is just \(q\).

So the entire expression simplifies to:

\[
q \wedge p
\]

### Conclusion:
Thus, the original expression simplifies to \(q \wedge p\). Therefore, \((\neg p \vee q) \wedge (p \wedge (\neg p \vee q))\) is logically equivalent to \(q \wedge p\).

Would you like further details or have any questions?

Here are some related questions:
1. What is the distributive law in logic?
2. How does the distributive law apply to conjunctions and disjunctions?
3. What is the significance of contradictions like \(p \wedge \neg p\) in logic?
4. How can logical identities be used to simplify complex expressions?
5. What are common pitfalls when simplifying logical expressions?

**Tip:** When simplifying logical expressions, always look for contradictions like \(p \wedge \neg p\) or tautologies like \(p \vee \neg p\), as they can drastically simplify the expression.

Explore the proof of the logical equivalence (~p ∨ q) ∧ (p ∧ (~p ∨ q)) using logical identities and distributive laws. This problem delves into complex logical expressions and their simplification, suitable for advanced high school and college-level students.

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