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:

Correct Option

Neither 1 nor 2

Let's analyze each statement:

Statement 1:

The given conditions are 2x(S)3y and 3x(T)4z.
Translating the symbols:
S means 'not less than', which implies 2x >= 3y.
T means 'equal to', which implies 3x = 4z.
From 3x = 4z, we can express x as x = (4z)/3.
Substitute this value of x into the first inequality:
2 * (4z)/3 >= 3y
8z/3 >= 3y
Multiplying both sides by 3 yields 8z >= 9y.
The statement to be evaluated is 9y(P)8z.
P means 'greater than', so this translates to 9y > 8z.
Our derivation 8z >= 9y contradicts the condition 9y > 8z. Therefore, Statement 1 is incorrect.

Statement 2:

The given conditions are x(Q)2y and y(R)z.
Translating the symbols:
Q means 'less than', which implies x < 2y.
R means 'not greater than', which implies y <= z.
From y <= z, multiplying both sides by 2 gives 2y <= 2z.
Combining the inequalities x < 2y and 2y <= 2z, we deduce x < 2y <= 2z.
This chain of inequalities implies x < 2z.
The statement to be evaluated is x(R)z.
R means 'not greater than', so this translates to x <= z.
From x < 2z, it is not possible to definitively conclude x <= z. For example, if x = 1.5 and z = 1, then x < 2z (1.5 < 2) holds true, but x <= z (1.5 <= 1) is false. Therefore, Statement 2 is incorrect.

Incorrect Options:

Options 1, 2, and 3 are incorrect because both Statement 1 and Statement 2 have been demonstrated to be false based on the given definitions and logical derivations.