We are tasked with showing that the expression $-(p \land q) \lor (p \land q)$ simplifies to $\neg p$. To do this, we'll use some standard logical laws, including De Morgan's Law, Distributive Law, and Absorption Law. Let's break it down step by step.

Expression:

$-(p \land q) \lor (p \land q)$

Step 1: Apply De Morgan's Law

De Morgan's law tells us that: $\neg (p \land q) = \neg p \lor \neg q$ So, applying this law to the negation of $p \land q$, we get: $(\neg p \lor \neg q) \lor (p \land q)$

Step 2: Distribute $\neg p \lor \neg q$ over $(p \land q)$

Now, apply the Distributive Law: $(\neg p \lor \neg q) \lor (p \land q) = (\neg p \lor (p \land q)) \lor (\neg q \lor (p \land q))$ We will handle both parts separately.

Step 3: Simplify using Absorption Law

Let’s simplify $\neg p \lor (p \land q)$ first. By Absorption Law, we know: $\neg p \lor (p \land q) = \neg p$ This is because if $\neg p$ is true, the entire disjunction is true regardless of $p \land q$; and if $\neg p$ is false, $p \land q$ would be false too.

Now let’s simplify $\neg q \lor (p \land q)$. By another application of Absorption Law: $\neg q \lor (p \land q) = p$ This is because if $p \land q$ is true, the whole disjunction is true regardless of $\neg q$, and if $p \land q$ is false, then the truth of $\neg q$ determines the value of the disjunction.

Step 4: Simplified expression

After simplification, the expression becomes: $\neg p \lor p$

Step 5: Apply the Law of Excluded Middle

The Law of Excluded Middle tells us that: $\neg p \lor p = \text{True}$ This is always true since either $p$ or $\neg p$ will hold in every situation.

Step 6: Simplification

However, this is not yet fully aligned with $\neg p$. We must have misinterpreted some intermediate simplifications.

Can you clarify what