Let's analyze each option to determine the correct solution:
Option A: not (x == y)
This statement is logically equivalent to x != y. In Python, x != y checks whether two values x and y are not equal. Similarly, the statement not (x == y) checks if x and y are not equal too. Therefore, if x does NOT equal y, both will return true. This indicates they are logically equivalent.
Option B: x > y and x < y
This statement can never be true because x cannot be both greater and smaller than y at the same time. This does not equate to x != y.
Option C: x > y or x < y
This statement will be true if x is not equal to y. However, it's not completely equivalent to x != y, because it would return false if x equals y, while x != y also returns false in this case, signaling that they are not equivalent in all cases.
Option D: x >= y or x <= y
This statement checks if x is either greater than or equal to y, or if it is less than or equal to y. This will always return true, irrespective of whether x is equal to y or not. Hence, it is not equivalent to x != y either.
In conclusion, the statement that is logically equivalent to x != y is not (x == y), hence the answer is option A.