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?

Correct Option: (d) Neither 1 nor 2

To evaluate Statement 1: The symbols are defined as P for '>', T for '=', and S for 'not less than' (which means '≥'). Given: 2x(S)3y ⇒ 2x ≥ 3y, and 3x(T)4z ⇒ 3x = 4z ⇒ x = 4z/3. Substituting x = 4z/3 into the first inequality: 2(4z/3) ≥ 3y ⇒ 8z/3 ≥ 3y ⇒ 8z ≥ 9y ⇒ 9y ≤ 8z. The statement concludes 9y(P)8z, which means 9y > 8z. Since we derived 9y ≤ 8z, Statement 1 is incorrect. To evaluate Statement 2: Q is defined as '<' and R is defined as 'not greater than' (which means '≤'). Given: x(Q)2y ⇒ x < 2y, and y(R)z ⇒ y ≤ z. We need to check if x(R)z, which means x ≤ z. Let us test with values: Let z = 1. Since y ≤ z, let y = 1. Since x < 2y, we can choose x = 1.5 (as 1.5 < 2 is true). But here, x > z (since 1.5 > 1), which contradicts x ≤ z. Thus, Statement 2 is also incorrect. Therefore, neither Statement 1 nor Statement 2 is correct, making option (d) the correct choice.