Animated Solution for Mathematics - Sets and Relations: If the truth value of the statement $(P\wedge (\sim R))\to ((\sim R)\wedge Q)$ is $F$, then the truth value of which of the following is $F$ ?

My dear student, welcome to the beautiful world of mathematical logic. Today, we are not just solving a problem; we are embarking on a journey to understand the very structure of truth.

We are given a logical statement: $(P\wedge (\sim R))\to ((\sim R)\wedge Q)$, and we are told that its truth value is False. This is our starting point, our anchor.

In logic, the implication operator $\to$ is like a gatekeeper. It only allows a False result under one specific condition.

If we view our statement as $X\to Y$, where $X$ is $(P\wedge (\sim R))$ and $Y$ is $((\sim R)\wedge Q)$, we must remember the golden rule: an implication is False if and only if the premise $X$ is True and the conclusion $Y$ is False.

This is the only way to break the promise of an implication. So, we immediately know:

Now, let us play detective. We have two clues. First, $(P\wedge (\sim R))=T$.

For an 'AND' ($\wedge$) operation to be True, both components must be True. This forces $P=T$ and $\sim R=T$.

If the negation of $R$ is True, then $R$ itself must be False. We have already found two of our three variables: $P$ is True and $R$ is False.

Next, we look at our second clue: $((\sim R)\wedge Q)=F$. We already know $\sim R$ is True.

So, our equation simplifies to $(T\wedge Q)=F$. For an 'AND' operation to result in False when one input is already True, the other input must be False.

Therefore, $Q$ must be False. We have successfully unraveled the mystery: $P=T$, $Q=F$, and $R=F$.

Now that we have our golden keys, we must test the options to see which one yields a False result. Let us examine Option (D): $\sim (R\vee Q)\to \sim P$.

Substituting our values, we get $\sim (F\vee F)\to \sim T$. Inside the parenthesis, $F\vee F$ is False.

So, the expression becomes $\sim (F)\to \sim T$. Applying the negations, $\sim (F)$ becomes True, and $\sim T$ becomes False.

We are left with $T\to F$. As we established in our very first step, $T\to F$ is False.

We have found our answer!

Logic is not just about symbols; it is about clarity of thought. By breaking down the complex into the simple, we turned a daunting problem into a series of logical steps.

Keep practicing this method, and you will find that even the most complex problems become clear. You have the tools; now go forth and conquer!