Question

The ceiling function of a real number \( x \), denoted by \( ce(x) \), is defined as the smallest integer that is greater than or equal to \( x \). Similarly, the floor function, denoted by \( fl(x) \), is defined as the largest integer that is smaller than or equal to \( x \). Which one of the following statements is NOT correct for all possible values of \( x \)?

• A. \( ce(x) \geq x \)
• B. \( fl(x) \leq x \)
• C. \( ce(x) \geq fl(x) \)
• D. \( fl(x)<ce(x) \)

Hint:

Remember that the ceiling function always rounds up, while the floor function always rounds down. So, for non-integer values, \( fl(x) \) will always be less than \( ce(x) \).

Answer 1

The Correct Option is D

The ceiling function \( ce(x) \) provides the smallest integer that is greater than or equal to \( x \).
The floor function \( fl(x) \) provides the largest integer that is smaller than or equal to \( x \).
We will now examine each statement:
Option (A): \( ce(x) \geq x \).
This statement is accurate because \( ce(x) \) is defined as the smallest integer that is greater than or equal to \( x \).
Option (B): \( fl(x) \leq x \).
This statement is accurate because \( fl(x) \) is defined as the largest integer that is smaller than or equal to \( x \).
Option (C): \( ce(x) \geq fl(x) \).
This statement is accurate because the ceiling of any number \( x \) will always be greater than or equal to its floor.
Option (D): \( fl(x) < ce(x) \).
This statement is NOT universally true. Consider the case where \( x \) is an integer. In such a scenario, \( fl(x) = x \) and \( ce(x) = x \), meaning \( fl(x) \) is equal to \( ce(x) \), not strictly less than it.
Therefore, the incorrect statement is option (D).