Q. If P means ‘greater than (>)’; Q means less than (<)’; R means ‘not greater than (>)’; S means ‘not less than (<)’ and T means ‘equal to (=)’, then consider the following statements:

# 1. If 2x(S)3y and 3x(T)4z, then 9y(P)8z.

2. If x(Q)2y and y(R)z, then x(R)z.

Which of the statements given above is/are correct?

a) 1 only

b) 2 only

c) Both 1 and 2

d) Neither 1 nor 2

Correct answer: d) Neither 1 nor 2

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

Let’s analyze each statement step by step:

## Statement 1: If 2x(S)3y and 3x(T)4z, then 9y(P)8z.

1. 2x(S)3y means 2x is not less than 3y, or 2x ≥ 3y

2. 3x(T)4z means 3x is equal to 4z, or 3x = 4z

3. From (2), we can say x = 4z/3

4. Substituting this in (1): 2(4z/3) ≥ 3y

5. Simplifying: 8z/3 ≥ 3y

6. Multiplying both sides by 3: 8z ≥ 9y

7. This is equivalent to 9y ≤ 8z, which is the opposite of 9y(P)8z (9y > 8z)

Therefore, statement 1 is incorrect.

### Statement 2: If x(Q)2y and y(R)z, then x(R)z.

1. x(Q)2y means x < 2y

2. y(R)z means y is not greater than z, or y ≤ z

3. From (1) and (2), we can say x < 2y and y ≤ z

4. This means x < 2z

5. However, x(R)z means x is not greater than z, or x ≤ z

6. While x < 2z is true, we cannot conclude that x ≤ z is always true

Therefore, statement 2 is also incorrect.

Since both statements are incorrect, the answer is: d) Neither 1 nor 2