Q76. 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?

Let's translate the given symbols: P: > Q: < R: ≤ (not greater than) S: ≥ (not less than) T: = **Statement 1: If 2x(S)3y and 3x(T)4z, then 9y(P)8z.** Translate: If 2x ≥ 3y and 3x = 4z, then 9y > 8z. From 3x = 4z, we can express x in terms of z: x = 4z/3.

Substitute this into the first inequality: 2(4z/3) ≥ 3y 8z/3 ≥ 3y Multiply by 3: 8z ≥ 9y. The conclusion given is 9y > 8z. This contradicts our derived inequality (8z ≥ 9y), as 9y cannot be strictly greater than 8z if 8z is greater than or equal to 9y.

Thus, Statement 1 is incorrect. **Statement 2: If x(Q)2y and y(R)z, then x(R)z.** Translate: If x < 2y and y ≤ z, then x ≤ z. From y ≤ z, we can infer that 2y ≤ 2z. We have two inequalities: x < 2y and 2y ≤ 2z.

Combining these, we get x < 2z. The conclusion given is x ≤ z. However, x < 2z does not necessarily imply x ≤ z. For example, if x=3 and z=2, then x < 2z (3 < 4) is true, but x ≤ z (3 ≤ 2) is false.

Thus, Statement 2 is incorrect. Therefore, neither Statement 1 nor Statement 2 is correct. This question tests logical reasoning with inequalities and symbolic representation.