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

Imagine you are standing at the threshold of a digital circuit, where every signal is either a high voltage (True) or a low voltage (False). This is the world of Boolean Algebra, the language of computers and the bedrock of logical reasoning.

Today, we are going to demystify the expression $(p\wedge \sim q)\Rightarrow (q\vee \sim p)$. It might look like a jumble of symbols, but by the end of this journey, you will see it as a simple, elegant logical statement.

The Truth Table

When you face a logical expression, the most robust tool in your arsenal is the Truth Table. Think of it as a map of all possible realities.

Since we have two variables, $p$ and $q$, there are exactly $2^2=4$ possible scenarios. We list them systematically: (T, T), (T, F), (F, T), and (F, F).

By breaking the expression into its components—first the negations $\sim p$ and $\sim q$, then the conjunction $(p\wedge \sim q)$, and finally the disjunction $(q\vee \sim p)$—we can evaluate the entire structure.

The key is to remember that the implication $A\Rightarrow B$ is only False when $A$ is True and $B$ is False. When we compute this for our expression, we find that the result is False only in the second row.

Logical Identities

Now, let's look at the 'soul' of the problem. There is a powerful identity in logic: $A\Rightarrow B\equiv \sim A\vee B$.

If we apply this to our expression $(p\wedge \sim q)\Rightarrow (q\vee \sim p)$, we get:

Applying De Morgan's Law to the first part, $\sim (p\wedge \sim q)$ transforms into $(\sim p\vee q)$. Now our expression looks like:

Because the OR operator is associative and commutative, we can rearrange this to $(\sim p\vee \sim p)\vee (q\vee q)$, which simplifies to $\sim p\vee q$. Note that $\sim p\vee q$ is exactly the definition of $p\Rightarrow q$.

By comparing our truth table results with the truth table for $p\Rightarrow q$, we see they are identical. We have successfully navigated the complexity and arrived at the core truth.

This problem isn't just about symbols; it's about understanding how conditions and outcomes interact. Whether you use the systematic truth table or the elegant path of logical identities, you are learning to think like a mathematician.

The expression $(p\wedge \sim q)\Rightarrow (q\vee \sim p)$ is logically equivalent to $p\Rightarrow q$. Keep practicing, and soon, these logical gates will feel as natural as breathing.