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

To find the negation of the logical statement $(p\vee q)\wedge (q\vee (\sim r))$, we first represent the entire expression as a single entity. We apply the negation operator $\sim$ to the whole statement:

In logic, when we negate a conjunction (an 'AND' statement), the operator transforms into a disjunction (an 'OR' statement). This is the fundamental bridge we must cross to begin our simplification.

We invoke De Morgan's Law, which states that $\sim (A\wedge B)\equiv (\sim A)\vee (\sim B)$. Applying this to our expression, the negation distributes to both brackets, and the central conjunction flips:

By breaking the primary connection, we have successfully reduced the complexity of the expression into two manageable parts.

Next, we apply De Morgan's Law again to the internal disjunctions. The negation of $(p\vee q)$ becomes $(\sim p)\wedge (\sim q)$, and the negation of $(q\vee (\sim r))$ becomes $(\sim q)\wedge (\sim (\sim r))$.

The expression now stands as:

Note how the internal 'OR' operators have flipped to 'AND' operators. This transformation is essential for further simplification.

We apply the Double Negation Law, where $\sim (\sim r)\equiv r$. Substituting this into our expression, we obtain:

Observe that the term $\sim q$ is common to both parts of the disjunction. We can now apply the Distributive Law in reverse to factor out the common term:

By navigating the logical operators and applying the laws of Boolean Algebra, we have arrived at the final simplified expression:

This result confirms the logical structure of the problem and aligns with the expected solution.