Proving Floor and Ceiling of a Rational Number

Question

Suppose x,y $ \in \mathbb{Z}^+ $

Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$

I was considering using the definition of floor and ceiling to prove this. But this does not seem like a valid proof to me as I assume the right hand side is already equal to the left hand side.

This is what I have so far:

Let n$ \in \mathbb{Z} $

$\lceil x/y \rceil = n \iff n-1 < x/y \leq n $

$\lfloor (x-1)/y \rfloor = n -1 \iff n-1 \leq (x-1)/y < n $
Then $\lfloor (x-1)/y \rfloor +1 = n \iff n-1 \leq (x-1)/y < n + 1 $

Then both sides are equivalent.

Would a better way be to prove using a contradiction by assuming both sides are not equal?

Answer 1

If $y$ divides $x$ then:

• $\lceil\frac{x}{y}\rceil=\frac{x}{y}$
• $\lfloor\frac{x-1}{y}\rfloor=\frac{x}{y}-1$

Hence $\lceil\frac{x}{y}\rceil=\lfloor\frac{x-1}{y}\rfloor+1$

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If $y$ does not divide $x$ then:

• $\lceil\frac{x}{y}\rceil=\lfloor\frac{x}{y}\rfloor+1$
• $\lfloor\frac{x-1}{y}\rfloor=\lfloor\frac{x}{y}\rfloor$

Hence $\lceil\frac{x}{y}\rceil=\lfloor\frac{x-1}{y}\rfloor+1$

Answer 2

starting from line 2, the inequality can be rewritten as

\begin{equation*} n-1+1/y \leq x/y < n+1/y \end{equation*} since x,y are integers, the left inequality is equivalent to

\begin{equation*} n-1 < x/y \end{equation*}

and the right inequality is equivalent to \begin{equation*} x/y \leq n \end{equation*}

Answer 3

I might want to use a property of integer arithmetic $a \lt b \iff a+1 \le b$, as in:

$\qquad\;\;\lceil x/y \rceil = n $

$ \iff n-1 \lt x/y \leq n $

$ \iff (n-1)y \lt x \leq ny $

$\iff (n-1)y \le x-1 \lt ny$

$\iff n-1 \le (x-1)/y \lt n $

$\iff \lfloor(x-1)/y\rfloor =n-1$

$\iff \lfloor(x-1)/y\rfloor +1=n$

so $\lceil x/y \rceil = \lfloor(x-1)/y\rfloor$ + 1