Let's go through the step-by-step simplification table to ensure clarity and correctness in each step:

1. Original Expression:

$[(m \to \neg n) \vee (m \wedge n)] \wedge p$ This is the starting point, and we aim to simplify this expression.

2. Apply the Implication Rule:

The implication $m \to \neg n$ can be rewritten using the rule $m \to \neg n = \neg m \vee \neg n$: $[(\neg m \vee \neg n) \vee (m \wedge n)] \wedge p$ This is the correct transformation from the original implication.

3. Apply the Associative Law:

We now combine the terms using the associative law of disjunction. The expression becomes: $(\neg m \vee \neg n \vee (m \wedge n)) \wedge p$ We don't do much simplifying yet here; this step is primarily about grouping the terms.

4. Apply the Distributive Law:

Next, we distribute $\neg m \vee \neg n$ over the conjunction $m \wedge n$. This step involves expanding: $(\neg m \vee \neg n \vee m) \wedge (\neg m \vee \neg n \vee n) \wedge p$ Here, we break up the disjunction $\neg m \vee \neg n \vee (m \wedge n)$ into two parts as a distributive step. The logic is:

• $(\neg m \vee \neg n \vee m)$ represents the first part, and
• $(\neg m \vee \neg n \vee n)$ represents the second part.

5. Apply the Complement Law:

Now, we apply the complement law. We know that:

• $\neg m \vee m = \text{True}$,
• $\neg n \vee n = \text{True}$.

Thus, the expression simplifies to: $\text{True} \wedge \text{True} \wedge p$

6. Apply the Identity Law:

Finally, applying the identity law, $\text{True} \wedge p = p$, the expression simplifies to: $p$

Final Result:

The simplified expression is $p$.

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Would you like further details or have any questions on the steps?

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Here are five related questions you might find interesting:

1. How does the distributive