Ceiling and Floor Functions Explained

This question explores the properties of the ceiling function, denoted as $ce(x)$, and the floor function, denoted as $fl(x)$, for any real number $x$.

• Ceiling Function ($ce(x)$): Defined as the smallest integer that is greater than or equal to $x$.
• Floor Function ($fl(x)$): Defined as the largest integer that is smaller than or equal to $x$.

We need to identify the statement among the given options that is NOT always true for all real numbers $x$.

Analyzing Option 1: $ce(x) \ge x$

By the definition of the ceiling function, $ce(x)$ is the smallest integer that is greater than or equal to $x$. This directly implies that $ce(x)$ must always be greater than or equal to $x$.

• Example 1: If $x = 4.7$, then $ce(4.7)$ is the smallest integer greater than or equal to 4.7, which is 5. Here, $5 \ge 4.7$.
• Example 2: If $x = 6$ (an integer), then $ce(6)$ is the smallest integer greater than or equal to 6, which is 6. Here, $6 \ge 6$.

Therefore, the statement $ce(x) \ge x$ is always correct.

Analyzing Option 2: $fl(x) \le x$

Similarly, the floor function $fl(x)$ is defined as the largest integer that is smaller than or equal to $x$. This means $fl(x)$ must always be less than or equal to $x$.

• Example 1: If $x = 4.7$, then $fl(4.7)$ is the largest integer smaller than or equal to 4.7, which is 4. Here, $4 \le 4.7$.
• Example 2: If $x = 6$ (an integer), then $fl(6)$ is the largest integer smaller than or equal to 6, which is 6. Here, $6 \le 6$.

Therefore, the statement $fl(x) \le x$ is always correct.

Analyzing Option 3: $ce(x) \ge fl(x)$

We know that $ce(x)$ is an integer $\ge x$, and $fl(x)$ is an integer $\le x$. Since $ce(x)$ is greater than or equal to $x$, and $fl(x)$ is less than or equal to $x$, it follows logically that $ce(x)$ must always be greater than or equal to $fl(x)$.

• Example 1: If $x = 4.7$, then $ce(4.7) = 5$ and $fl(4.7) = 4$. Clearly, $5 \ge 4$.
• Example 2: If $x = 6$ (an integer), then $ce(6) = 6$ and $fl(6) = 6$. Here, $6 \ge 6$.

Therefore, the statement $ce(x) \ge fl(x)$ is always correct.

Analyzing Option 4: $fl(x) < ce(x)$

This statement claims that the floor value is strictly less than the ceiling value. Let's consider two cases:

• Case 1: $x$ is NOT an integer.
  If $x$ is not an integer, like $x = 4.7$, then $fl(4.7) = 4$ and $ce(4.7) = 5$. In this case, $4 < 5$, so the statement $fl(x) < ce(x)$ holds true.
• Case 2: $x$ IS an integer.
  If $x$ is an integer, like $x = 6$, then by definition, $fl(6) = 6$ and $ce(6) = 6$. In this situation, $fl(x) = ce(x)$. The statement $fl(x) < ce(x)$ becomes $6 < 6$, which is false.

Since the statement $fl(x) < ce(x)$ is not true when $x$ is an integer, it is NOT correct for all possible values of $x$.

Conclusion

Based on the analysis, the statement that is NOT correct for all possible values of $x$ is $fl(x) < ce(x)$. This inequality fails specifically when $x$ is an integer, as in that case, $fl(x) = ce(x)$.