Prove Relations: $e,b,d\in \mathbb{Z},d\neq 0$

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Discussion Overview

The discussion revolves around the mathematical definitions and properties of the division operation denoted as $e \text{ div } d$ for integers $e$ and $d$, where $d \neq 0$. Participants explore how to prove specific relations involving the floor and ceiling functions based on the sign of $d$, and seek clarification on definitions and conventions used in different contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that if $d > 0$, then $e \text{ div } d = \lfloor \frac{e}{d} \rfloor$, while if $d < 0$, then $e \text{ div } d = \lceil \frac{e}{d} \rceil$.
  • Another participant questions the definitions of $e \text{ div } d$, $\lfloor x \rfloor$, and $\lceil x \rceil$, seeking clarity on their meanings.
  • Some participants discuss the relationship between $e$, $d$, and the remainder $r$ in the equations $e = k \cdot d + r$, where $k$ is defined as $e \text{ div } d$.
  • Concerns are raised about the lack of a universally accepted definition for $e \text{ div } d$, suggesting that definitions may vary by context, such as mathematics versus computer science.
  • There is a call for more precise definitions and clarification on the differences between the two cases for positive and negative $d$, as well as the restrictions on the remainder $r$.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and interpretations of $e \text{ div } d$, with no consensus reached on a single definition or approach. The discussion remains unresolved regarding the precise definitions and implications of the floor and ceiling functions in this context.

Contextual Notes

Participants note that the definitions of $e \text{ div } d$ may depend on the specific mathematical or computational context, and there are variations in how the floor and ceiling functions are interpreted, particularly for negative values.

evinda
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Hello! (Wave)Let $e,b \in \mathbb{Z}, d \neq 0$.
How could we prove the following? Could you maybe give me a hint?

  • If $d>0$ then $e \text{ div } d = \lfloor \frac{e}{d} \rfloor$
    $$$$
  • If $d<0$ then $e \text{ div } d = \lceil \frac{e}{d} \rceil $

Could we show the above, using the definitions? (Thinking)

$$\lfloor x \rfloor =max \{ m \in \mathbb{Z}: m \leq x \}$$

$$\lceil x \rceil=\min \{ l \in \mathbb{Z}: l \geq x\}$$
 
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What are the definitions of $e\text{ div }d$, $\lfloor x\rfloor$ and $\lceil x\rceil$?
 
Evgeny.Makarov said:
What are the definitions of $e\text{ div }d$, $\lfloor x\rfloor$ and $\lceil x\rceil$?

If $d>0$ it is $e=k \cdot d+r, r<d$ where $k=e\text{ div }d=\lfloor \frac{e}{d} \rfloor$, right? (Thinking)

And if $d<0$, is it then like that? (Thinking)

$e=k \cdot d+r, r<d$ where $k=e\text{ div }d=\lceil \frac{e}{d} \rceil$
 
evinda said:
If $d>0$ it is $e=k \cdot d+r, r<d$ where $k=e\text{ div }d=\lfloor \frac{e}{d} \rfloor$, right? (Thinking)

And if $d<0$, is it then like that? (Thinking)

$e=k \cdot d+r, r<d$ where $k=e\text{ div }d=\lceil \frac{e}{d} \rceil$
I don't know because $e\text{ div }d$ does not have a universally accepted definition. You have to go with the one used in your book or course. In contrast, $\lfloor x\rfloor$ is pretty unambiguous, but even then there are variations: for example, W|A rounds negative values up instead of down even though "integer part" is usually considered a synonym of "floor function".

Also, I assume that $\lfloor \frac{e}{d} \rfloor$ and $\lceil \frac{e}{d} \rceil$ are not part of the definition of $e\text{ div }d$. Then what is the difference between the two clauses for $d>0$ and $d<0$ in your post? And are there are no other restrictions on $r$, such as $0\le r$? In short, it would be nice if you wrote complete and precise definitions.
 
Here's some information on "a div b". I think in mathematics, method 2 or 3 below is used most often, while in CS, method 1 is usually the case.

9jfzas.png

xeg95z.png
 

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