Prove that the greatest integer function f:R→R, given f(x)=⌊x⌋ is neither one-one nor onto where ⌊x⌋ denotes the greatest integer function less than or equal to x.

The greatest integer function, denoted as f(x) = ⌊x⌋, maps real numbers to the largest integer less than or equal to x. To show that this function is neither one-one nor onto, we can analyze its properties.

Not One-One

A function is one-one (or injective) if different inputs produce different outputs. For f(x) = ⌊x⌋, consider the following examples:

• f(2.3) = ⌊2.3⌋ = 2
• f(2.9) = ⌊2.9⌋ = 2

Here, both 2.3 and 2.9 yield the same output of 2. Thus, f is not one-one since multiple inputs can lead to the same output.

Not Onto

A function is onto (or surjective) if every possible output in the codomain is achieved by some input from the domain. The codomain of f is the set of real numbers, but the outputs of f are only integers. For instance, there is no x in R such that f(x) = 1.5, as f(x) can only produce whole numbers. Therefore, f is not onto since not every real number can be represented as an output.

Summary

In conclusion, the greatest integer function f(x) = ⌊x⌋ is neither one-one nor onto because it maps multiple inputs to the same output and does not cover all real numbers in its range.