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Thread: Yet two more questions I don't understand.

  1. #1
    Jan 2010

    Yet two more questions I don't understand.

    Question 1:

    We have seen that congruence modulo m is an equivalence relation on Z(all integers) for any integer m >= 1. Describe the set of equivalence classes for congruence modulo 8 - how many distinct classes are there and what are they?

    What I am getting out of this is that there are 7 distinct classes and they are 1-7. But then again I am completely clueless on this modulo stuff and I've already gone to the professor with no luck on understanding it any better.

    Question 2:

    Show that addition modulo 8 is well-defined. (Use arbitrary elements from two equivalence classes [a] and [b] and how that the result is an element in [a] + [b]).

    We briefly went over this in class and I again didn't understand it because of the whole modulo thing. I think it is something to do with taking two classes that are equivalent to mod 8 (which I am not sure how to figure out) and then proving that if you add them together you get whatever the question is asking.

    That's just it though, I am not understanding the question nor how to find a part of what it is asking. Can anyone clarify this for me?

    Help is appreciated!
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  2. #2
    Mar 2009
    I'll give it a try Yaeger- and hopefully someone more senior on the board will fix up the misakes. I guess you know how the mod function works...
    We'll do it with mod 3 because that's easier:
    Notice how (2 mod 3) = (5 mod 3) = 2. Well, when you think about mod 3 as being a relation on the set of whole numbers, there are infinitely many numbers (every 3rd member of the Set to be exact) that are related to 2 by the mod 3 relation. So when you start to write down the members of the relationship, you get:

    \rho:= \{(0, 0), (1, 1), (2, 2), (3, 0), (4, 1), (5, 2), ..., (8, 2), ... (2 + 3n, 2), ...\}

    An equivalence class is a short-hand way of naming the three partitions of the relationship, that is all the members of the relationship that have either (x, 0), (y, 1) or (z, 3) in them, so $\displaystyle [1]_{mod 3} = [4]_{mod 3}$ are two equivalence classes which are congruent. Both of them are elements from the same representational system, or partition.

    Another example of equivalence classes are in the rational numbers, where $\displaystyle \frac{1}{2} = \frac{2}{4} \ldots$

    Question 2 just shows you how you can do some basic math stuff with them. Try it out.
    Last edited by bmp05; Feb 4th 2010 at 12:27 PM. Reason: latex/ spelling funkiness
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  3. #3
    Jan 2010
    Thanks for the post. I understand the equivalence class thing now. But I am still unsure about the second question. But I am currently trying to figure it out now.

    Thanks again!
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