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

Mastering Boolean Logic

Welcome, future engineers. Today, we are not just solving a problem; we are learning the language of logic itself. Boolean algebra is the bedrock of every computer, every circuit, and every algorithm you will ever design.

When you look at an expression like $\sim (p\vee q)\vee (\sim p\wedge q)$, do not see a jumble of symbols. See a puzzle waiting to be solved with elegance and precision.

De Morgan's Law

Our journey begins with the first term: $\sim (p\vee q)$. This is a classic setup for De Morgan's Law.

Imagine you are standing at a gate where you are told that '$p$ or $q$' is true. If you negate that entire condition, you are essentially saying that neither $p$ nor $q$ is allowed to be true.

Mathematically, this is the beauty of De Morgan's Law:

Notice how the 'or' symbol ($\vee$) flips upside down to become an 'and' symbol ($\wedge$). This is not just a rule; it is a fundamental shift in perspective. We have now transformed our expression into $(\sim p\wedge \sim q)\vee (\sim p\wedge q)$.

Now, look at the expression with fresh eyes. We have two distinct parts separated by an 'or' operator: $(\sim p\wedge \sim q)$ and $(\sim p\wedge q)$.

Do you see the symmetry? Both terms share a common element: $\sim p\wedge$.

Just as you would factor out an $x$ in a polynomial like $ax+ay=a(x+y)$, we can factor out $\sim p\wedge$ here. This is the Distributive Law in action. By pulling out the common factor, we rewrite the expression as:

Suddenly, the complexity collapses. We have isolated the core of the problem.

Now, focus entirely on the bracket: $(\sim q\vee q)$. This is the Complement Law.

Think about it logically: if you have a statement $q$, and you combine it with its own negation $\sim q$ using an 'or' operator, one of them MUST be true. It is impossible for both to be false.

Therefore, the statement $(\sim q\vee q)$ is always true, regardless of the value of $q$. In logic, we call this a Tautology, denoted by $T$. Our expression is now simply $\sim p\wedge T$.

We are at the finish line. We have $\sim p\wedge T$. Think of $T$ as the 'identity' element in logic—it is the neutral observer.

If you perform an 'and' operation between any statement and 'True', the truth value of the entire expression depends solely on the original statement. Thus:

We have bypassed the tedious, error-prone process of truth tables and arrived at the answer through pure, structural reasoning. This is the power of Boolean algebra. The final simplified result is $\sim p$.