I'm trying to prove that a function that I have found is injective. While I do know that I can do this by supposing that f(x) = f(x') and then doing some algebra to show that therefore x = x'. However, my function uses the floor function so I'm stuck with how to proceed.
Specifically I want to show that $f(x) = x + \lfloor{\frac{x-1}{2}}\rfloor$ where x is an integer and f(x) is also some integer. All I really need help with is somehow getting rid of the floor function. If I can get into this format $f(x) = x - n$ Then I'd be sweet. Could I just suppose that in order for the function to be equal that the two floor parts must be equal? or is that already considered to be assuming the conclusion that x = x?
Any help or pointers are appreciated. Please try to really dumb it down. :)
Your function is the sum of a strictly increasing function ($x$) with an increasing one ($\left\lfloor\frac{x-1}2\right\rfloor$). Therefore, it is a strictly increasing function, and every strictly increasing function is injective.