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?

The correct answer/option is Neither 1 nor 2.

Let us analyze the meaning of the operations first:
- P means "greater than" (>)
- Q means "less than" (<)
- R means "not greater than" (which means less than or equal to, i.e., ≤)
- S means "not less than" (which means greater than or equal to, i.e., ≥)
- T means "equal to" (=)

Analysis of Statement 1:
The statement is: "If 2x(S)3y and 3x(T)4z, then 9y(P)8z."
Using the symbol definitions, we can write the given conditions as:
1) 2x(S)3y ⇒ 2x ≥ 3y
2) 3x(T)4z ⇒ 3x = 4z ⇒ x = 4z/3

Let us substitute the value of x from (2) into (1):
$2\left(\frac{4z}{3}\right)\ge 3y$
Multiplying both sides by 3:
8z ≥ 9y
This can also be written as:
9y ≤ 8z (which is 9y(R)8z)

The statement concludes that 9y(P)8z, which means 9y > 8z. However, we derived that 9y ≤ 8z. Therefore, Statement 1 is incorrect.

Analysis of Statement 2:
The statement is: "If x(Q)2y and y(R)z, then x(R)z."
Using the symbol definitions, we can write the given conditions as:
1) x(Q)2y ⇒ x < 2y
2) y(R)z ⇒ y ≤ z

From these, we want to check if x(R)z (which means x ≤ z) is definitely true.
Let us choose values to test this:
If we let z = 1 and y = 1 (since y ≤ z is satisfied).
Then, from x < 2y, we get x < 2(1) ⇒ x < 2.
If we choose x = 1.5 (which is less than 2), then x = 1.5 and z = 1.
Here, x is greater than z (1.5 > 1), which means x ≤ z does not hold.
Since we found a case where x ≤ z is false, the conclusion x(R)z does not logically follow. Therefore, Statement 2 is also incorrect.

Since both Statement 1 and Statement 2 are incorrect, the correct option is "Neither 1 nor 2".