Is the statement [tex]$\lceil x\rceil=\lfloor x+1\rfloor$[/tex] true for all real numbers? Explain.
Answer (2)
Consider the case where x is an integer, then ⌈ x ⌉ = x and ⌊ x + 1 ⌋ = x , so the statement holds.
Consider the case where x is not an integer and can be written as x = n + f , where n is an integer and 0 < f < 1 .
Then ⌈ x ⌉ = n + 1 and ⌊ x + 1 ⌋ = ⌊ n + f + 1 ⌋ = n + 1 , so the statement holds.
Since the statement holds for both cases, the statement is true for all real numbers. T r u e
Explanation
Understanding the Problem We want to determine if the statement ⌈ x ⌉ = ⌊ x + 1 ⌋ is true for all real numbers x . Here, ⌈ x ⌉ represents the smallest integer greater than or equal to x , and ⌊ x ⌋ represents the largest integer less than or equal to x .
Considering Cases Let's consider two cases: when x is an integer and when x is not an integer.
Case 1: x is an Integer Case 1: If x is an integer, then ⌈ x ⌉ = x . Also, x + 1 is an integer, so ⌊ x + 1 ⌋ = x + 1 − 1 = x . Therefore, ⌈ x ⌉ = x = ⌊ x + 1 ⌋ .
Case 2: x is not an Integer Case 2: If x is not an integer, then we can write x = n + f , where n is an integer and 0 < f < 1 . Then ⌈ x ⌉ = n + 1 , since n is the greatest integer less than x , so the next integer is n + 1 . Also, x + 1 = n + f + 1 , so ⌊ x + 1 ⌋ = n + 1 because 0 < f < 1 , which means 1 < f + 1 < 2 , so n + 1 < n + f + 1 < n + 2 , and the greatest integer less than n + f + 1 is n + 1 . Thus, ⌈ x ⌉ = n + 1 = ⌊ x + 1 ⌋ .
Conclusion Since the statement is true for both cases, the statement ⌈ x ⌉ = ⌊ x + 1 ⌋ is true for all real numbers.
Examples
Imagine you're measuring fabric and need to round up to the nearest whole inch ( ⌈ x ⌉ ) for cutting, but your ruler starts an inch ahead. Checking ⌊ x + 1 ⌋ gives you the same result, ensuring you have enough fabric. This principle applies in scenarios where you need consistent rounding, such as inventory management or resource allocation, ensuring accuracy regardless of the starting point.
The statement ⌈ x ⌉ = ⌊ x + 1 ⌋ is true for all real numbers. This is validated by considering cases where x is an integer and where it is not. In both scenarios, the equality holds, confirming the statement's truth.
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