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Thread: Equivalence relation/ Equivalence classes

  1. #1
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    Equivalence relation/ Equivalence classes

    I'v got difficult quenstion:

    A={0,1,2,3,4,5} group of the numbers 0,1,2,3,4,5. find the appropriate equivalence relation to the next equivalence classes(marked by A/R):

    I) A/R={ {1,2},{3},{4,0,5} }

    II) A/R={ {0,1,2,3,4,5} }

    III) A/R={ {0,1,2} , {3,4,5} }

    help me please! I have to submit the answer that question in the next few hours
    good day,
    Haim
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  2. #2
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    Re: Equivalence relation/ Equivalence classes

    Is this about group theory? If so, you need to find cosets.
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  3. #3
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    Re: Equivalence relation/ Equivalence classes

    Quote Originally Posted by haim View Post
    I'v got difficult quenstion:
    A={0,1,2,3,4,5} SET of the numbers 0,1,2,3,4,5. find the appropriate equivalence relation to the next equivalence classes(marked by A/R):
    I) A/R={ {1,2},{3},{4,0,5} }
    II) A/R={ {0,1,2,3,4,5} }
    III) A/R={ {0,1,2} , {3,4,5} }
    Quote Originally Posted by mekun View Post
    Is this about group theory? If so, you need to find cosets.
    @mekun
    Neither! It is about Equivalence relation/ Equivalence classes. Please read titles.

    @haim, your translator used group where the correct term is set.

    If $A/\mathcal{R}=\{B,C,D,E,\cdots\}$ is a set of non-empty, pairwise disjoint subsets of $A$ then the Equivalence relation
    $\mathcal{R}=\{B\times B,~C\times C,~D\times D,~E\times E\cdots\}$
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  4. #4
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    Re: Equivalence relation/ Equivalence classes

    Quote Originally Posted by haim View Post
    I'v got difficult quenstion:

    A={0,1,2,3,4,5} group of the numbers 0,1,2,3,4,5. find the appropriate equivalence relation to the next equivalence classes(marked by A/R):

    I) A/R={ {1,2},{3},{4,0,5} }
    Do you not know what "equivalence class" means? The fact that one equivalence class is {1, 2} tells you that 1 is equivalent to 2 (and, of course, each is equivalent to itself). The fact that one equivalence class is {3} tells you that 3 is equivalent to itself and not equivalent to any other member of this set. The fact that one equivalence set is {4, 0, 5} tells you that each of 4, 0, 5 is equivalent to itself and each of the other 2.

    In terms of a "set of ordered pairs", the equivalence relation is {(0, 0), (1, 1), (2, 2), (3,3), (4,4), (5, 5), {1, 2), (2, 1), (0, 4), (4, 0), (0, 5), (5, 0), (4, 5), (5, 4)}.

    II) A/R={ {0,1,2,3,4,5} }
    Since this is a single set, each is equivalent to all the others. As a set of ordered pairs, it contains all 6^2= 36 possible ordered pairs.

    III) A/R={ {0,1,2} , {3,4,5} }
    You should be able to do this your self. In include all 9 possible ordered pairs from {0, 1, 2} and all 9 possible ordered pairs from {3, 4, 5} for a total of 18 ordered pairs.

    help me please! I have to submit the answer that question in the next few hours
    good day,
    Haim
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  5. #5
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    Re: Equivalence relation/ Equivalence classes

    thank you all! My friend helped me too with those question, and the answers are like yours (verified already). it's simply that I have no experience with that subject at all so i couldn't answer on almost any question. I'm sorry if there was any misunderstanding with my question, it's just that I translated the question from Hebrew, so me and the dictionary are not familiar with the appropriate technical definitions.
    thanks again!
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