Proof: For all real numbers x and y if x + floor of (x) = y + floor of (y) then x = y

Question

I am trying to determine if this statement is true or false:

I think that it is true, if i let x = 2.5 then the left side is 4.5 and if i let y be anything but 2.5 then x + floor of (x) cant not equal y + floor of (y).

Since n<= x < n + 1 any number y between n and n + 1 will have the same floor as x but if x is not y the statement cant be true.

How would i start to prove this?

Answer 1

Let $f(x)=x+floor(x)$. You question is equivalent to asking whether $f$ is one-to-one, or whether there are any horizontal lines that intersect the graph of $f$ more than once. So, what does the graph look like?

Answer 2

Recall that $\lfloor t+n\rfloor=\lfloor t\rfloor +n$ for all $t\in\mathbb{R}$ and $n\in\mathbb{Z}$. If $x$ and $y$ are real numbers such that $$x+\lfloor x\rfloor= y+\lfloor y\rfloor\,,$$ then taking the floor function on both sides, we get $$2\,\lfloor x\rfloor=\big\lfloor x+\lfloor x\rfloor\big\rfloor=\big\lfloor y+\lfloor y\rfloor\big\rfloor=2\,\lfloor y\rfloor\,.$$ Thus, $\lfloor x\rfloor=\lfloor y\rfloor$. The rest should be trivial.