Animated Solution for Mathematics - Sets and Relations: Consider the following three statements : P : 5 is a prime number. Q : 7 is a factor of 192. R : L.C.M. of 5 and 7 is 35. Then the truth value of which one of the following statements is true ?

A Logical Journey

Welcome, fellow explorer of the mathematical universe. Today, we are not just solving a problem; we are building a foundation in the elegant, binary world of mathematical logic.

This JEE-style problem is a classic, designed to test your precision and your ability to remain calm under the pressure of multiple, interconnected statements. Let us embark on this journey together.

Deconstructing the Statements

Before we can build a logical structure, we must understand our building blocks. We have three statements: $P$, $Q$, and $R$.

Think of this as verifying the integrity of your materials before you start construction.

Statement $P$ claims that $5$ is a prime number. By definition, a prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself.

Since the only factors of $5$ are $1$ and $5$, $P$ is undeniably True, or $P\equiv T$.

Next, we examine Statement $Q$, which claims that $7$ is a factor of $192$. To verify this, we perform a simple division:

Because this is not an integer, $7$ cannot be a factor of $192$. Thus, $Q$ is False, or $Q\equiv F$.

Finally, Statement $R$ states that the Least Common Multiple (LCM) of $5$ and $7$ is $35$. Since $5$ and $7$ are both prime numbers, they are coprime, meaning their only common factor is $1$.

For any two coprime numbers, their LCM is simply their product: $5\times 7=35$. Therefore, $R$ is True, or $R\equiv T$.

The Logic Toolkit

Now that we have our truth values—$P\equiv T$, $Q\equiv F$, and $R\equiv T$—we are ready to test the options. In logic, we use three primary operators: the AND operator ($\wedge$), the OR operator ($\vee$), and the NOT operator ($ eg$).

Remember, the AND operator is a strict gatekeeper; it only returns True if both inputs are True. The OR operator is inclusive; it returns True if at least one input is True. The NOT operator is the great inverter, turning True to False and False to True.

The Systematic Evaluation

We must now evaluate each option to see which one results in a final truth value of True. Let us proceed with the discipline of a scientist.

For Option 1, we have $(P \land Q) \lor ( eg R)$. Substituting our values, we get $(T \land F) \lor ( eg T)$.

Since $T\wedge F$ is False and $ eg T$ is False, we have $F\vee F$, which is False.

For Option 2, we have $( eg P) \land ( eg Q \land R)$. Substituting, we get $( eg T) \land ( eg F \land T)$.

This simplifies to $F\wedge (T\wedge T)$, which is $F\wedge T$, resulting in False.

For Option 3, we have $( eg P) \lor (Q \land R)$. Substituting, we get $( eg T) \lor (F \land T)$.

This simplifies to $F\vee F$, which is False.

Finally, we arrive at Option 4: $P \lor ( eg Q \land R)$. Substituting our values, we get $T \lor ( eg F \land T)$.

Inside the parentheses, $ eg F$ is True, so we have $T\wedge T$, which is True. The expression becomes $T\vee T$, which is True.

We have systematically dismantled the problem and found our answer. Option 4 is the only statement that evaluates to True.

The key takeaway here is not just the answer, but the process. By breaking down complex statements into their atomic truth values and applying logical operators with care, you can solve any problem in this domain.