A Complete Set of Representatives

[adanedhel728]

I've got an assignment that asks me to give a "complete set of representatives for R/~," and I've spent a long time trying to figure out what that means with no luck at all. I've got the definition of "complete set of representatives" right here and I don't understand it even a little bit. I've tried googling the phrase, also with no luck. Here's the definition the book gives:

Let ~ be an equivalence relation on a set A. A subset of A containing exactly one element from each equivalence class is called complete set of representatives of A/~.

Then it gives an example,

For example, {0,1,...,n-1} and {1,2,...,n} are both complete sets of representatives of Z modulo n. In general, if B is a complete set of representatives of A/~, then A/~={[x]|x is a member of B}

(I made the text big because it's kind of hard to see all the brackets and stuff in the normal size.)

Any help would be greatly appreciated. All I need is to figure out what that definition means, because it makes no sense to me at all. And the examples are not helpful.

Let me know if there's any more information I need to give. In case anyone's familiar with it, I'm taking this from a book called Doing Mathematics: An Introduction to Proofs and Problem Solving by Steven Galovich.

Thanks,
Andrew

 

[pka]

I've got an assignment that asks me to give a "complete set of representatives for R/~," All I need is to figure out what that definition means, because it makes no sense to me at all. I'm taking this from a book called Doing Mathematics: An Introduction to Proofs and Problem Solving by Steven Galovich.

Do not over think this process. It is easy to understand using a simple example.
Consider the set if digits: \(\displaystyle \mathbb{D} = \left\{ {0,1,2,3,4,5,6,7,8,9} \right\}\).
On that define an equivalence relation: \(\displaystyle a\Re b\) if and only if \(\displaystyle a\;\& \;b\;\) have the same remainder when divided by 3.
Now there are exactly three equivalence classes: \(\displaystyle \left\{ {0,3,6,9} \right\}\;,\;\left\{ {1,4,7} \right\}\;\& \;\left\{ {2,5,8} \right\}\).
Suppose that I take one digit from each class. Example: \(\displaystyle \left\{ {3,7,5} \right\}\).
That is a complete set of representatives for \(\displaystyle \mathbb{D}/\Re\).
Of course, a more ‘natural’ complete set of representatives may be \(\displaystyle \left\{ {0,1,2} \right\}\).

I hope what I have written matches Galovich’s text. I cannot find my copy (I wrote a negative review of his effort).

If you still have problems, then post an exact exercise from that textbook. We will try to help.

 

[adanedhel728]

Alright, I think I understand the concept now. The specific problem that I have is written this way--

[attachment=1:3unwbst5]define the set.jpg[/attachment:3unwbst5]
So this is the answer I have now--

[attachment=0:3unwbst5]answer.jpg[/attachment:3unwbst5]
I kind of had a hard time with this, but my reasoning is that when x=0, there's a representative of integers, and when x is anything other than zero, then x is a representation of any integer plus whatever that real number less than one is, so that should cover all equivalence classes. Does that sound about right?

 

[pka]

When x=0, there's a representative of integers, and when x is anything other than zero, then x is a representation of any integer plus whatever that real number less than one is, so that should cover all equivalence classes.

That is correct and very well put. Good for you.

 

[adanedhel728]

Ah! Thanks very much, this has been very helpful.

--Andrew