Question

(Discrete Math)

See the picture in the attachment.

How do I get started with this problem?

Answer 1

Recall the definition of the floor function: if
`n` is an integer and
`n\le x<n+1`, then
`\lfloor x\rfloor=n`. So
`\lfloor4x\rfloor=n` iff
`\frac n4\le x<\frac{n+1}4`.

So let's consider four case.

First, suppose
`n\le x<n+\frac14` where
`n\in\mathbb Z`. It follows that


`\lfloor x\rfloor=\left\lfloor x+\frac14\right\rfloor=\left\lfloor x+\frac12\right\rfloor=\left\lfloor x+\frac34\right\rfloor=n`

because at most, we have


`n\le x<n+\frac14\implies n+\frac34\le x+\frac34<n+1\implies\left\lfloor x+\frac34\right\rfloor=n`

Meanwhile,
`\lfloor4x\rfloor=4n`, which follows immediately from


`n\le x<n+\frac14\implies 4n\le4x<4n+1`

and so


`\lfloor4x\rfloor=4n=n+n+n+n=\lfloor x\rfloor+\left\lfloor x+\frac14\right\rfloor+\left\lfloor x+\frac12\right\rfloor+\left\lfloor x+\frac34\right\rfloor`

Then for the second case, you can consider what happens when you have
`n+\frac14\le x<n+\frac12`; for the third,
`n+\frac12\le x<n+\frac34`; and for the fourth,
`n+\frac34\le x<n+1`.