The Greatest and Least Integer Functions


Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

Verified step by step guidance

1. Understand the definitions: The floor function ⌊x⌋ gives the greatest integer less than or equal to x, while the ceiling function ⌈x⌉ gives the smallest integer greater than or equal to x.
2. Consider a real number x that is not an integer, such as 2.5. For this value, ⌊2.5⌋ = 2 and ⌈2.5⌉ = 3. Clearly, ⌊x⌋ ≠ ⌈x⌉ in this case.
3. Now consider a real number x that is an integer, such as 3. For this value, ⌊3⌋ = 3 and ⌈3⌉ = 3. Here, ⌊x⌋ = ⌈x⌉.
4. From these observations, we can conclude that ⌊x⌋ = ⌈x⌉ if and only if x is an integer.
5. Therefore, ⌊x⌋ = ⌈x⌉ for all real x is not true; it only holds when x is an integer.

Key Concepts

Greatest Integer Function (Floor Function)

The greatest integer function, denoted as ⌊x⌋, returns the largest integer less than or equal to x. For example, ⌊3.7⌋ equals 3, while ⌊-2.3⌋ equals -3. This function effectively 'rounds down' any real number to the nearest integer.

Least Integer Function (Ceiling Function)

The least integer function, denoted as ⌈x⌉, returns the smallest integer greater than or equal to x. For instance, ⌈3.7⌉ equals 4, and ⌈-2.3⌉ equals -2. This function 'rounds up' any real number to the nearest integer.

Comparison of Floor and Ceiling Functions

The question asks whether ⌊x⌋ equals ⌈x⌉ for all real x. In general, these two functions yield different results unless x is already an integer. For non-integer values, ⌊x⌋ will be less than ⌈x⌉, highlighting that they are not equal for all real numbers.