Animated Solution for Mathematics - Sets and Relations: The negation of the expression $q\vee ((\sim q)\wedge p)$ is equivalent to

We are tasked with simplifying the symbolic logic expression:

To visualize this, imagine a universal set $U$ containing two overlapping sets, $p$ and $q$. We first analyze the inner component: $((\sim q)\wedge p)$.

In set theory, this represents the intersection of the complement of $q$ and the set $p$. Physically, this is the region inside $p$ that lies strictly outside $q$.

Next, we introduce the union operator $\vee$ with $q$. We are combining the "only $p$" region with the entire circle $q$.

By painting the entire circle $q$ and adding the remaining part of $p$ not already covered, we effectively shade the entire area covered by both $p$ and $q$. Thus, the expression $q\vee ((\sim q)\wedge p)$ is logically equivalent to $p\vee q$.

The problem requires the negation of this entire expression, which is $\sim (p\vee q)$. Visually, this represents the region of the universal set that belongs to neither $p$ nor $q$.

To express this mathematically, we invoke De Morgan's Law, which states that the negation of a union is the intersection of the negations:

Applying this to our simplified expression, we obtain:

To ensure absolute rigor, we apply algebraic laws directly to the original expression:

Applying De Morgan's Law to the outer bracket, we get:

Applying De Morgan's Law again to the inner bracket yields:

Using the Distributive Law to expand this expression:

Since $(\sim q)\wedge q$ is a contradiction (always False, denoted as $F$), the expression simplifies to:

Because $F$ combined with any statement via an 'OR' operator leaves the other term unchanged, we arrive at the final result: $(\sim p)\wedge (\sim q)$