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Math Help - [SOLVED] Equivalance Relations

  1. #1
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    Smile [SOLVED] Equivalance Relations

    Can somebody clarify me how equivalence classes are obtained ?
    And how can I answer this type of Qn ?

    Qn:- Let S={1,2,3,4,5} have a partition consisting of the sets {1,3,5} and {2,4} .Show that this partition determines an equivalence relation .
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  2. #2
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    An equivalence relation always forms a partition on the set in which it is defined. Each cell, or element, of the partition is an equivalence class.

    For example, in your set:

    {1,2,3,4,5} you have the partition
    {1,3,5}
    {2,4}

    A good choice would be the equivalence relation:
    element a of A is related to element b of A, if:
    a is of the same parity as b
    notice {1, 3, 5} are all odd,
    {2,4} are both even.

    I'll leave it up to you to confirm that this equivalence relation is in fact an equivalence relation:

    It's reflexive, a R a
    Symmetric, a R b implies b R a
    Transitive.. a R b and b R c implies a R c....

    a R b is the conventional notation used to denote element a is Related to b...
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  3. #3
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    Also. Given an equivalence relation, an equivalence class is formally defined and denoted as the following:

    Given an equivalence relation defined on some set A

    [a] : {b ϵ A | b R a}

    That is, the set of all the elements of A which are related to a.

    As an example, consider the equivalence class of nonnegative integers congruent to 1 modulo 2:

    [1(mod 2)] = {1, 3, 5, 7, 9, 11, etc} essentially, all the odd integers.

    similarly the equivalence class of integers congruent to 0 mod 2:
    [0 (mod 2)] = {0, 2, 4, 6, 8, 10, etch} all the even nonnegative integers.

    The relation a R b if a is congruent to b modulo 2 is another equivalence relation that might help you with your question. The only equivalence classes of this relation are [0] and [1]. Together, these two equivalence classes partition all of the integers in the set on which this relation is defined.
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  4. #4
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    But I wouldn't directly copy this. I'm a bit rusty on my notation.
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  5. #5
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    Equivalence Relations

    Hello K A D C Dilshan
    Quote Originally Posted by K A D C Dilshan View Post
    Can somebody clarify me how equivalence classes are obtained ?
    And how can I answer this type of Qn ?

    Qn:- Let S={1,2,3,4,5} have a partition consisting of the sets {1,3,5} and {2,4} .Show that this partition determines an equivalence relation .
    You might find it helpful to study the examples and my replies in this posting.

    Grandad
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  6. #6
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    Thumbs up Thank you !!!!!

    Quote Originally Posted by dr3amrunn3r View Post
    Also. Given an equivalence relation, an equivalence class is formally defined and denoted as the following:

    Given an equivalence relation defined on some set A

    [a] : {b ϵ A | b R a}

    That is, the set of all the elements of A which are related to a.

    As an example, consider the equivalence class of nonnegative integers congruent to 1 modulo 2:

    [1(mod 2)] = {1, 3, 5, 7, 9, 11, etc} essentially, all the odd integers.

    similarly the equivalence class of integers congruent to 0 mod 2:
    [0 (mod 2)] = {0, 2, 4, 6, 8, 10, etch} all the even nonnegative integers.

    The relation a R b if a is congruent to b modulo 2 is another equivalence relation that might help you with your question. The only equivalence classes of this relation are [0] and [1]. Together, these two equivalence classes partition all of the integers in the set on which this relation is defined.
    Actually dr3amrunn3r I had this idea before .
    Your definition of equivalence classes is not correct I think .
    I think it should be corrected as follows
    [a] = {x : (a,x) ∊ R} Isn`t it ?
    so [1(mod2)] and [2(mod2)] are the partitions as I thought as we can not take 0 .
    But I had no experience on building relations using partitions .
    I think a is congruent to b modulo 2 is something similar to a≣b(mod2) i.e 2 | (a-b)
    Thank you very much for confirming the solution .
    If I have done any mistake please tell me .
    Last edited by K A D C Dilshan; April 10th 2009 at 03:39 AM.
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