Problem 18 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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? \(\quad\) (a) \((\sim \mathrm{P}) \vee(\mathrm{Q} \wedge \mathrm{R})\) (b) \((P \wedge Q) \vee(\sim R)\) (c) \((\sim \mathrm{P}) \wedge(\sim \mathrm{Q} \wedge \mathrm{R})\) (d) \(\mathrm{P} \vee(\sim \mathrm{Q} \wedge \mathrm{R})\)

Step-by-Step Solution

Answer

The true statement is (d) \( P \vee(\sim Q \wedge R) \).

Step 1: Identify Truth Values of Individual Statements

Assess the truth values of the individual statements:- Statement \( P \): '5 is a prime number.' This is true because 5 is only divisible by 1 and itself.- Statement \( Q \): '7 is a factor of 192.' This is false because when you divide 192 by 7, it does not result in an integer.- Statement \( R \): 'The L.C.M. of 5 and 7 is 35.' This is true because 5 and 7 are both primes, and their least common multiple is their product, which is 35.

Step 2: Evaluate Each Option

Now, evaluate each compound statement with the truth values of \( P \), \( Q \), and \( R \):(a) \( \sim P \vee (Q \wedge R) \): Since \( P \) is true, \( \sim P \) is false. \( Q \) is false, making \( Q \wedge R \) false. Therefore, this option is false.(b) \( (P \wedge Q) \vee \sim R \): \( P \wedge Q \) is false because \( Q \) is false. \( R \) is true, making \( \sim R \) false. This option is false.(c) \( \sim P \wedge (\sim Q \wedge R) \): \( \sim P \) is false. \( \sim Q \wedge R \) is true because \( \sim Q \) is true (since \( Q \) is false) and \( R \) is true. However, `false AND anything` is false, so this option is false.(d) \( P \vee (\sim Q \wedge R) \) : \( P \) is true. No need to evaluate further because true OR anything is true.

Key Concepts

Truth Values

In logical reasoning, truth values are fundamental. They indicate whether statements are true or false. Truth values help us analyze and evaluate complex logical expressions.
Understanding individual statement truth values allows for accurate computation of more complex propositions. The notion is simple:

• True: The statement is correct.
• False: The statement is incorrect.

Let's consider an example based on three statements:

• Statement P: '5 is a prime number.' Since 5 can only be divided by 1 and itself, this is a true statement.
• Statement Q: '7 is a factor of 192.' 7 does not divide 192 evenly, making this statement false.
• Statement R: 'The L.C.M. of 5 and 7 is 35.' Since the least common multiple of two distinct primes is their product, this statement is true.

Mastering truth values assists in 'computing' the correctness of compound statements in logical expressions.

Logical Connectives

Logical connectives are the glue that allows us to combine individual statements into compound propositions. The primary logical connectives include AND (\( \wedge \)), OR (\( \vee \)), and NOT (\( \sim \)).
Combining the earlier truth values, you can derive the whole expression truthfulness. Here's a short guide:

• AND (\( \wedge \)): True only if both components are true.
• OR (\( \vee \)): True if at least one component is true.
• NOT (\( \sim \)): Flips the truth value, turning true to false and vice versa.

For example, in statement (b): - \( (P \wedge Q) \vee (\sim R) \)
- \(P \wedge Q\) is false as \(Q\) is false, and \(\sim R\) is false as \(R\) is true.
Thus, the full expression is false.
Understanding these connectives is pivotal to proper logical expression evaluation and problem-solving.

Least Common Multiple

Least common multiple (LCM) is the smallest positive integer divisible by a set of numbers. Calculating LCM is a key skill in arithmetic, often invoked in problems involving repeated patterns or common denominators.
When dealing with numbers like 5 and 7:

• Since they are both prime numbers, the LCM is found by multiplying them.
• Thus, LCM(5, 7) = 35.

Why does this matter in logical reasoning? Well, in the given exercise, deriving the truth value of statement \( R \) depends heavily on understanding the concept of LCM.
Mastery of LCM helps in:

• Simplifying fractions and ratios.
• Finding timing of cyclical events.
• Applying logical conditions effectively in math reasoning problems.

By connecting arithmetic skills with logical reasoning, students enhance their overall problem-solving capabilities.