# 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

**Explanation : **

**Statement 1: **“If 2x is not less than 3y AND 3x equals 4z, then 9y is greater than 8z”

Analysis:

1. From 3x equals 4z, we can say x equals 4z/3

2. Putting this in first condition: 2(4z/3) is not less than 3y

3. Simplifying: 8z/3 is not less than 3y

4. Multiplying both sides by 3: 8z is not less than 9y

5. Therefore: 9y is less than or equal to 8z

6. But statement claims 9y is greater than 8z

7. This is a contradiction

### Statement 2: “If x is less than 2y AND y is not greater than z, then x is not greater than z”

Analysis:

1. From first condition: x < 2y

2. From second condition: y ≤ z

3. Putting these together: x < 2z

4. However, x < 2z does not guarantee that x ≤ z

5. Counter example: Let z = 1, y = 1, x = 1.9

– Here x < 2y is true (1.9 < 2)

– y ≤ z is true (1 ≤ 1)

– But x ≤ z is false (1.9 is not ≤ 1)

Therefore, both statements are false.