Question

The ceiling function of a real number \(x\), denoted by \(ce(x)\), is defined as the smallest integer \(\geq x\). The floor function, denoted by \(fl(x)\), is defined as the largest integer \(\leq x\). 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:

Floor and ceiling are equal if and only if \(x\) is an integer; otherwise, floor is strictly less than ceiling.

Answer 1

The Correct Option is D

Step 1: Recall definitions.
- Floor function: \(fl(x) = \max\{n \in \mathbb{Z} \mid n \leq x\}\). - Ceiling function: \(ce(x) = \min\{n \in \mathbb{Z} \mid n \geq x\}\). Step 2: Verify each statement. \begin{itemize} \item (A) \(ce(x) \geq x\): Always true, since ceiling is defined as the smallest integer not less than \(x\). \item (B) \(fl(x) \leq x\): Always true, since floor is defined as the largest integer not greater than \(x\). \item (C) \(ce(x) \geq fl(x)\): Always true, because ceiling is never less than floor. \item (D) \(fl(x) < ce(x)\): Not always true. For integer values of \(x\), we have \(fl(x) = ce(x) = x\). Hence the strict inequality fails. \end{itemize} Final Answer: \[ \boxed{fl(x) < ce(x)} \]