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

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In summary, if $d>0$, then $e=k \cdot d+r, r<d$, where $k=e\text{ div }d$. If $d<0$, then $e=k \cdot d+r, r<d$, where $k=e\text{ div }d=\lceil \frac{e}{d} \rceil$.
  • #1
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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  • #2
What are the definitions of $e\text{ div }d$, $\lfloor x\rfloor$ and $\lceil x\rceil$?
 
  • #3
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 $k=e\text{ div }d=\lceil \frac{e}{d} \rceil$
 
  • #4
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.
 
  • #5
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.

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

What is a relation?

A relation is a set of ordered pairs that relate elements from one set, called the domain, to elements in another set, called the range. In other words, a relation shows how one set of values is connected to another set of values.

How do you prove a relation?

To prove a relation, you must show that for every element in the domain, there is at least one corresponding element in the range that satisfies the given rule or condition. This can be done by providing a specific example, using a table, or using algebraic equations.

What does it mean for e, b, and d to be in the set of integers?

The set of integers, denoted as ℝ, is a collection of whole numbers and their negative counterparts. This means that all three variables, e, b, and d, can take on any positive or negative whole number value.

Can d be equal to 0 in this relation?

No, the given condition states that d must not equal 0. This is because division by 0 is undefined in mathematics and can lead to invalid solutions.

What is the significance of e, b, and d being in the same relation?

The fact that all three variables are in the same relation means that there is a connection or relationship between them. This could be in the form of a mathematical equation, a pattern, or a common property that they all share.

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