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.