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 \( ce(x) \) returns the smallest integer greater than or equal to \( x \).
The floor function \( fl(x) \) returns the largest integer smaller than or equal to \( x \).
Now, we analyze each option:
Option (A): \( ce(x) \geq x \).
This is true since the ceiling of \( x \) is the smallest integer greater than or equal to \( x \).
Option (B): \( fl(x) \leq x \).
This is true since the floor of \( x \) is the largest integer smaller than or equal to \( x \).
Option (C): \( ce(x) \geq fl(x) \).
This is true because the ceiling of \( x \) is always greater than or equal to the floor of \( x \).
Option (D): \( fl(x) < ce(x) \).
This is NOT true for all \( x \). For example, if \( x \) is an integer, then \( fl(x) = ce(x) = x \), so \( fl(x) \) is not strictly less than \( ce(x) \).
Thus, the correct answer is option (D).