Animated Solution for Mathematics - Sets and Relations: If $P⟹(q\vee r)$ is false, then the truth values of p, q, r are respectively :-

Imagine you are a detective, and you have been handed a case involving a mysterious logical expression: $P⟹(q\vee r)$. You are told that this entire statement is False.

Your mission is to uncover the truth values of $P$, $q$, and $r$. This is not just a math problem; it is a journey into the very foundation of how we reason.

The Implication Operator

The primary operator here is the implication, denoted by $⟹$. Think of it as a strict judge.

In the world of Boolean logic, an implication $A⟹B$ is like a promise. It is only broken—meaning it evaluates to False—in one specific, unforgiving scenario: when the first part ($A$) is True, but the second part ($B$) is False.

If the first part is False, the promise is considered 'vacuously true', and the whole statement is True. If both are True, the promise is kept, and the statement is True.

Therefore, for our expression $P⟹(q\vee r)$ to be False, we have no choice but to conclude that $P$ must be True, and the entire bracketed expression $(q\vee r)$ must be False. This is our first major breakthrough. We have already solved for $P$! We know with absolute certainty that $P=T$.

Solving for $q$ and $r$

Now, we turn our attention to the second part of our mystery: $(q\vee r)=\text{False}$. The operator here is the disjunction, or the logical 'OR' ($\vee$).

The 'OR' operator is much more generous than the implication. It is True if even one of its inputs is True. It only demands that both inputs be False for the entire statement to be False.

It is a team effort. If $q$ were True, the whole bracket would be True. If $r$ were True, the whole bracket would be True.

The only way for the team to fail—to be False—is if both $q$ and $r$ are False. Thus, we have our final pieces of the puzzle: $q=F$ and $r=F$.

By methodically breaking down the expression, we have uncovered the truth: $P=T$, $q=F$, and $r=F$.

The sequence is $(T,F,F)$. It is elegant, it is precise, and it is the only possible solution.

Logic is the language of the universe, and by mastering these simple rules, you are learning to speak it fluently. Keep practicing, keep questioning, and remember that every complex problem is just a collection of simple truths waiting to be revealed.