Animated Solution for Mathematics - Sets and Relations: The compound statement $(P\vee Q)\wedge (\sim P)\Rightarrow Q$ is equivalent to:

A Journey into Logical Reasoning

Welcome, future engineers! Today, we are not just solving a problem; we are peeling back the layers of logical structure. Mathematical reasoning is the bedrock of computer science and complex engineering.

When you look at a statement like $(P\vee Q)\wedge (\sim P)\Rightarrow Q$, it might look like a jumble of symbols, but I want you to see it as a precision-engineered machine. Our goal is to understand how this machine behaves under every possible condition.

The Systematic Search

Whenever you face a logical expression with two variables, $P$ and $Q$, you are dealing with a system that has exactly $2^2=4$ possible states. These states are $(T,T),(T,F),(F,T),$ and $(F,F)$. Think of these as the four possible configurations of a circuit.

To understand the behavior of our expression, we must test it against all four. We start by evaluating the disjunction $P\vee Q$. This is the 'inclusive OR'—it is true if at least one of the inputs is true.

So, in our table, we get $T,T,T,F$. Now, we introduce the negation $\sim P$. This is the inverter; it flips the truth value of $P$. If $P$ is True, $\sim P$ is False, and vice versa. Our column for $\sim P$ becomes $F,F,T,T$.

The Conjunction and the Implication

Now, let's combine these using the conjunction $\wedge$ (the AND operator). Remember, an AND gate is strict; it only outputs True if both inputs are True.

Looking at our columns for $(P\vee Q)$ and $\sim P$, we find that only in the third row are both conditions met. Thus, the result for $(P\vee Q)\wedge (\sim P)$ is $F,F,T,F$. This is the 'antecedent' of our final implication.

Now, we reach the heart of the problem: the implication $\Rightarrow$. The rule is simple but crucial: $A\Rightarrow B$ is false only when $A$ is True and $B$ is False.

Let's test our antecedent against $Q$. In every row where our antecedent is True, $Q$ is also True. We never encounter a $T\Rightarrow F$ scenario. Consequently, the entire statement evaluates to True in every single case. We have discovered a Tautology!

Finding the Equivalent

Since our original statement is a tautology, we are looking for an option that is also a tautology. Let's examine Option 4: $\sim (P\Rightarrow Q)\iff (P\wedge \sim Q)$.

This is a biconditional statement. We know that $P\Rightarrow Q$ is logically equivalent to $\sim P\vee Q$. If we negate this, $\sim (P\Rightarrow Q)$, we get $\sim (\sim P\vee Q)$, which, by De Morgan's laws, becomes $P\wedge \sim Q$.

So, the left side is equivalent to the right side. A statement of the form $X\iff X$ is always true. It is a tautology! Because both our original expression and Option 4 are tautologies, they are logically equivalent. You have successfully navigated the logic gate!