To determine whether the statement [tex]\(\lceil x \rceil = \lfloor x + 1 \rfloor \)[/tex] holds true for all real numbers, let's analyze it step-by-step with multiple test cases.
### Definitions
- The ceiling function [tex]\(\lceil x \rceil\)[/tex] gives the smallest integer greater than or equal to [tex]\(x\)[/tex].
- The floor function [tex]\(\lfloor x \rfloor\)[/tex] gives the largest integer less than or equal to [tex]\(x\)[/tex].
### Checking Various Examples
1. For [tex]\(x = -2.5\)[/tex]:
- [tex]\(\lceil -2.5 \rceil = -2\)[/tex]
- [tex]\(\lfloor -2.5 + 1 \rfloor = \lfloor -1.5 \rfloor = -2\)[/tex]
- Result: [tex]\(-2 = -2\)[/tex] (True)
2. For [tex]\(x = -1.1\)[/tex]:
- [tex]\(\lceil -1.1 \rceil = -1\)[/tex]
- [tex]\(\lfloor -1.1 + 1 \rfloor = \lfloor -0.1 \rfloor = -1\)[/tex]
- Result: [tex]\(-1 = -1\)[/tex] (True)
3. For [tex]\(x = 0\)[/tex]:
- [tex]\(\lceil 0 \rceil = 0\)[/tex]
- [tex]\(\lfloor 0 + 1 \rfloor = \lfloor 1 \rfloor = 1\)[/tex]
- Result: [tex]\(0 \neq 1\)[/tex] (False)
4. For [tex]\(x = 0.5\)[/tex]:
- [tex]\(\lceil 0.5 \rceil = 1\)[/tex]
- [tex]\(\lfloor 0.5 + 1 \rfloor = \lfloor 1.5 \rfloor = 1\)[/tex]
- Result: [tex]\(1 = 1\)[/tex] (True)
5. For [tex]\(x = 1.99\)[/tex]:
- [tex]\(\lceil 1.99 \rceil = 2\)[/tex]
- [tex]\(\lfloor 1.99 + 1 \rfloor = \lfloor 2.99 \rfloor = 2\)[/tex]
- Result: [tex]\(2 = 2\)[/tex] (True)
6. For [tex]\(x = 2\)[/tex]:
- [tex]\(\lceil 2 \rceil = 2\)[/tex]
- [tex]\(\lfloor 2 + 1 \rfloor = \lfloor 3 \rfloor = 3\)[/tex]
- Result: [tex]\(2 \neq 3\)[/tex] (False)
### Conclusion
From these examples, we observe that the statement [tex]\(\lceil x \rceil = \lfloor x + 1 \rfloor\)[/tex] is true for some values of [tex]\(x\)[/tex] but not for others. Specifically, the statement is false for [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex], where the calculated results did not match.
Therefore, [tex]\(\lceil x \rceil = \lfloor x + 1 \rfloor\)[/tex] is not true for all real numbers.
