Consider the following: A + B means A is neither smaller nor equal to B . A - B means A is not greater than B . A × B means A is not smaller than B . A B means A is neither greater nor equal to B . A B means A is neither smaller nor greater than B . Statement: P × Q , P - T , T R , R S Conclusion-1: Q T Conclusion-2: S + Q Which one of the following is correct in respect of the above Statement and the Conclusions?

B. Only Conclusion-2 follows from the Statement.

Reasoning : Logical Reasoning & Analytical Ability

Q 71 / 325

UPSC CSE Prelims 2023

Consider the following:

$A + B$ means $A$ is neither smaller nor equal to $B$ .
$A - B$ means $A$ is not greater than $B$ .
$A \times B$ means $A$ is not smaller than $B$ .
$A \div B$ means $A$ is neither greater nor equal to $B$ .
$A \pm B$ means $A$ is neither smaller nor greater than $B$ .

Statement: $P \times Q$ , $P - T$ , $T \div R$ , $R \pm S$
Conclusion-1: $Q \pm T$
Conclusion-2: $S + Q$
Which one of the following is correct in respect of the above Statement and the Conclusions?

EXPLANATION

Correct Option (B)

The first step involves translating the given symbolic relationships into standard mathematical inequalities:

$A + B$ signifies $A > B$ (A is neither smaller nor equal to B).
$A - B$ signifies $A \leq B$ (A is not greater than B).
$A \times B$ signifies $A \geq B$ (A is not smaller than B).
$A \div B$ signifies $A < B$ (A is neither greater nor equal to B).
$A \pm B$ signifies $A = B$ (A is neither smaller nor greater than B).

Applying these translations to the provided statement:

$P \times Q$ translates to $P \geq Q$ .
$P - T$ translates to $P \leq T$ .
$T \div R$ translates to $T < R$ .
$R \pm S$ translates to $R = S$ .

Combining these individual inequalities yields the comprehensive relationship:

$Q \leq P \leq T < R = S$

Now, each conclusion is evaluated based on this combined relationship:

Conclusion-1: $Q \pm T$

This conclusion translates to $Q = T$ . From the derived combined statement, we have $Q \leq P \leq T$ , which implies $Q \leq T$ . However, this does not definitively establish that $Q$ must be equal to $T$ . For instance, if $Q = 1, P = 2, T = 3$ , the condition $Q \leq P \leq T$ holds, but $Q \neq T$ . Therefore, Conclusion-1 does not necessarily follow.

Conclusion-2: $S + Q$

This conclusion translates to $S > Q$ . From the combined statement $Q \leq P \leq T < R = S$ , we can deduce that $Q \leq T$ and $T < S$ . If $Q$ is less than or equal to $T$ , and $T$ is strictly less than $S$ , it logically follows that $Q$ must be strictly less than $S$ . Hence, $S > Q$ . Therefore, Conclusion-2 follows from the statement.

Based on this analysis, only Conclusion-2 is valid. Thus, Option (B) is the correct answer.

Incorrect Options:

Options (A), (C), and (D) are incorrect because they do not accurately reflect the validity of the conclusions derived from the given statement.

Option (A) asserts that only Conclusion-1 follows. This is incorrect because Conclusion-1 ( $Q = T$ ) cannot be definitively inferred from the established inequalities.
Option (C) claims that both Conclusion-1 and Conclusion-2 follow. This is incorrect as Conclusion-1 is not a valid deduction.
Option (D) states that neither Conclusion-1 nor Conclusion-2 follows. This is incorrect because Conclusion-2 ( $S > Q$ ) is a direct and valid consequence of the combined statement.