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-   -   R.D Silverman's number theory homework (https://www.mersenneforum.org/showthread.php?t=14901)

 science_man_88 2011-01-19 00:31

R.D Silverman's number theory homework

I know most of you have agreed to ignore me and a few others, But I have a question about something I read doing the homework Silverman suggested.

[QUOTE]A binary relation on a set S is a subset R of S ×S. Usually, one writes
a ∼ b to mean that (a, b) ∈ R, where ∼ is some appropriate symbol, and
rather than refer to the relation as R, one refers to it as ∼[/QUOTE]

so ~ is just any random relation ? like x<y and y<z implies x<z or am I off target already, if so I'm going to be a long way off track by the time I get through the preliminaries.

 CRGreathouse 2011-01-19 00:51

[QUOTE=science_man_88;247266]so ~ is just any random relation ?[/QUOTE]

Right, just like x is just any random number.

 science_man_88 2011-01-19 00:53

I think I'm a bit off track as this reminds me of the later phrasing:

[QUOTE]A binary relation ∼ on a set S is called an equivalence relation if for
all x, y, z ∈ S, we have
• x ∼ x (reﬂexive property),
• x ∼ y implies y ∼ x (symmetric property), and
• [B]x ∼ y and y ∼ z implies x ∼ z (transitive property).[/B][/QUOTE]. I think I have a bit of this in my head and not enough of the basics like Silverman wants. can anyone help with me understanding the first equation or is that outside the Silverman law.

 science_man_88 2011-01-19 01:11

[QUOTE=CRGreathouse;247274]Right, just like x is just any random number.[/QUOTE]

Thank you very much for all the help you've provided on this forum (Especially this type of help).

 science_man_88 2011-01-19 01:51

[QUOTE]If ∼ is an equivalence relation on S, then for x ∈ S one deﬁnes the set
[x] := {y ∈ S : x ∼ y}[/QUOTE]

I don't see why they use the word on to describe it yet but I might soon ( maybe it's so it doesn't get confused with [TEX]\in[/TEX] (meaning in ?). My next line of questioning is on the equation part, Does this mean for all [TEX]x \in S[/TEX] there's a [TEX]y \in S[/TEX] such that the relation on S always holds ? I'll be back around 8-9 am (UTC -4:00 time).

 jasonp 2011-01-19 01:59

When I read that symbol I mentally translate it to 'a member of'.

 davieddy 2011-01-19 03:31

[QUOTE=jasonp;247301]When I read that symbol I mentally translate it to 'a member of'.[/QUOTE]
Who was reputed to have said "Whenever I hear the word cultere,
I reach for my revolver"?

Goering?

PS Don't mention the war.

 CRGreathouse 2011-01-19 05:30

[QUOTE=science_man_88;247299]I don't see why they use the word on to describe it yet but I might soon ( maybe it's so it doesn't get confused with [TEX]\in[/TEX] (meaning in ?). My next line of questioning is on the equation part, Does this mean for all [TEX]x \in S[/TEX] there's a [TEX]y \in S[/TEX] such that the relation on S always holds ?[/QUOTE]

No. For every x you find all y such that x is related to y (x ~ y) and call the set of all such y's "[x]".

Suppose I define ~ over the real numbers as "x ~ y if and only if x - y is an integer". Then 0.2 ~ 1.2, 0.2 ~ 2.2, 0.2 ~ 3.2, etc., so that [0.2] = {..., 0.2, 1.2, 2.2, ...}.

Is this ~ an equivalence relation? (Check if it has the three properties.)

 CRGreathouse 2011-01-19 05:39

[QUOTE=davieddy;247313]Who was reputed to have said "Whenever I hear the word cultere,
I reach for my revolver"?[/QUOTE]

[url]http://en.wikiquote.org/wiki/List_of_misquotations#revolver[/url]

 xilman 2011-01-19 07:29

[QUOTE=science_man_88;247282]Thank you very much for all the help you've provided on this forum (Especially this type of help).[/QUOTE]You're welcome. As Bob wrote earlier, you will find people here more than willing to help you if you play by the rules: show that you have made an honest effort; explain where you are having difficulty; ask questions in a polite manner.

I feel I ought to head off a possible misconception which, if you hold it, will cause confusion later. When CRGreathouse used the phrase "x is any random number" he clearly used "random" in the colloquial sense. However, it also has a precise mathematical meaning and you should be careful of using "random" in a context where you do not intend the technical meaning. In particular, the phrase "x is any number" has exactly the same intended meaning and does not suffer from the potentially misleading ambiguity.

Paul

 Raman 2011-01-19 07:35

@science_man_88: What is your educational qualification?

You said that you cannot afford to buy book. What is your father, mother?
What place are you from?

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