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Thread: Equivalence Relation

  1. #1
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    Equivalence Relation

    Give an example of an equivalence relation R on a set A = {1,2,3,4,5,6,7} with P the set of equivalence classes such that the following four properties are satisfied:
    1. |P|=3
    2. There exists no set S in P such that |S|=3
    3. 3 R 4 but 3 R 5
    4. there exists a set T in P such that 1, 7 in T

    So far the only relations elements in the relation I can think of is:

    R = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7)
    (3,5), (5,3), (1,7), (7,1)}

    I'm not sure how to get the rest... I seem to be missing a key part of how I am supposed to do this.
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  2. #2
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    Re: Equivalence Relation

    In general, you are looking for a partition of the elements of the original set into three sets, no set has exactly 3 elements, one sets contains both 3 and 5, one set (not necessarily different) contains 1,7. 4 is not in the set containing 3. Example:

    $$\{1,2,3,5,7\}, \{4\}, \{6\}$$

    This is a valid partition that satisfies all of the conditions. See if you can describe the relation that generates this partition.
    Last edited by SlipEternal; Oct 5th 2018 at 08:35 AM.
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  3. #3
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    Re: Equivalence Relation

    Quote Originally Posted by MrJank View Post
    Give an example of an equivalence relation R on a set A = {1,2,3,4,5,6,7} with P the set of equivalence classes such that the following four properties are satisfied:
    1. |P|=3
    2. There exists no set S in P such that |S|=3
    3. 3 R 4 but 3 R 5
    4. there exists a set T in P such that 1, 7 in T

    So far the only relations elements in the relation I can think of is:

    R = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7), (3,5), (5,3), (1,7), (7,1)}
    I'm not sure how to get the rest... I seem to be missing a key part of how I am supposed to do this.
    If we use $\mathcal{R}_x$ to stand the equivalence class determined by $x$ then that relation has five classes not three.
    $\mathcal{R}_1=\{1,7\}$,$\mathcal{R}_2=\{2\}$,$ \mathcal {R}_3=\{3,5\}$,$\mathcal{R}_4= \{4\}$,$\mathcal {R}_6= \{6\}$
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  4. #4
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    Re: Equivalence Relation

    R = {(1,1),(1,2),(1,3),(1,5),(1,7),(4,4),(6,6)}

    Is that close?
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  5. #5
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    Re: Equivalence Relation

    Quote Originally Posted by MrJank View Post
    R = {(1,1),(1,2),(1,3),(1,5),(1,7),(4,4),(6,6)}
    Is that close?
    That is not even an equivalence relation!
    Say the classes are: $\mathcal{R}_1=\{1,7\},\mathcal{R}_2=\{2,3,5,6\}, \mathcal {R}_4=\{4\}$[/QUOTE]

    Is that collection a partition of the set $\{1,2,3,4,5,6,7\}~?$
    If it is what is the corresponding equivalence relation?
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