Understanding Ceiling and Floor Functions

The ceiling function, denoted as $ce(x)$, gives the smallest integer that is greater than or equal to $x$. The floor function, denoted as $fl(x)$, gives the largest integer that is smaller than or equal to $x$. We need to find the statement that is not true for all real numbers $x$.

Analyzing Function Properties

• Statement 1: $ce(x) \ge x$
  By definition, the ceiling function $ce(x)$ is always an integer greater than or equal to $x$. This statement is always true. For instance, $ce(4.2) = 5$, and $5 \ge 4.2$. For integers, $ce(x) = x$, which also satisfies $ce(x) \ge x$.
• Statement 2: $fl(x) \le x$
  By definition, the floor function $fl(x)$ is always an integer smaller than or equal to $x$. This statement is always true. For example, $fl(4.2) = 4$, and $4 \le 4.2$. For integers, $fl(x) = x$, which also satisfies $fl(x) \le x$.
• Statement 3: $ce(x) \ge fl(x)$
  Since $fl(x)$ is the largest integer $\le x$ and $ce(x)$ is the smallest integer $\ge x$, it must be that $ce(x) \ge fl(x)$. If $x$ is an integer, $ce(x) = fl(x) = x$. If $x$ is not an integer, $fl(x) < x < ce(x)$, implying $ce(x) > fl(x)$. This statement is always true.
• Statement 4: $fl(x) < ce(x)$
  This statement claims the floor is always strictly less than the ceiling. However, consider when $x$ is an integer. For example, if $x=7$, then $fl(7) = 7$ and $ce(7) = 7$. In this specific case, $fl(x) = ce(x)$, which contradicts the statement $fl(x) < ce(x)$. Therefore, this statement is NOT correct for all possible values of $x$. It holds true only when $x$ is not an integer.

The statement that is NOT correct for all possible values of $x$ is $fl(x) < ce(x)$.