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Math Help - Question on equivalence relations and equivalence classes

  1. #1
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    Question on equivalence relations and equivalence classes

    Hi i need help with the following question

    Let X={1,2,3,4} and let A be the set X x X. Define a relation on A by
    (x,y)R(w,z)<=>x+y=w+z
    Show that R is an equivalence relation on A and describe (as subsets of A) the equivalence classes of R.


    The first bit involving that the statement is a equivalence relation, i manged to do the following however i am not sure if it is correct though.

    transitive since (x,y)R(x,y)=x+y=x+y
    symmetric since (w,z)R(x,y)=w+z=x+y
    transitive since (x,y)R(w,z) and let (w,z)R(u,v) ==> (x,y)R(u,v)=x+y=u+v

    However i have no idea on how to do the equivalence class stuff.

    thanks
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  2. #2
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    Quote Originally Posted by cooltowns View Post
    Let X={1,2,3,4} and let A be the set X x X. Define a relation on A by
    (x,y)R(w,z)<=>x+y=w+z
    Show that R is an equivalence relation on A and describe (as subsets of A) the equivalence classes of R.
    transitive since (x,y)R(x,y)=x+y=x+y reflexive?
    symmetric since (w,z)R(x,y)=w+z=x+y
    transitive since (x,y)R(w,z) and let (w,z)R(u,v) ==> (x,y)R(u,v)=x+y=u+v
    However i have no idea on how to do the equivalence class stuff.
    What pair(s) is(are) equivalence to (1,2)?
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  3. #3
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    yep sorry thats a mistake i was meant to say reflexive for (x,y)R(x,y)=x+y=x+y

    as i don't understand what you mean by what pairs are equivalent to (1,2) ?

    as in do you just mean that it is (2,1) since both (1,2) and (2,1) have the same elements hence equivalent ?
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  4. #4
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    The question is supposed to mean: what pairs are in relation R with (1,2)? Since R is an equivalence relation, one can say (maybe informally) that pairs related by R are equivalent.
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  5. #5
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    i am not sure on how to answer.

    what so specific about (1,2) what the other numbers 3 and 4 ?

    (1,2) i presume this means 1+2=1+2 if symmetric
    not sure on about symmetric and transitive relations though since if present it like (1,2)R(3,4) 1+2 does not equal 3+4

    i am just guessing here i am not really sure on how to approach the question though.
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  6. #6
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    Quote Originally Posted by cooltowns View Post
    i am not sure on how to answer.

    what so specific about (1,2) what the other numbers 3 and 4 ?

    (1,2) i presume this means 1+2=1+2 if symmetric
    not sure on about symmetric and transitive relations though since if present it like (1,2)R(3,4) 1+2 does not equal 3+4

    i am just guessing here i am not really sure on how to approach the question though.
    Is it true that (1,2)\mathcal{R}(2,1)?
    If so, then WHY?

    What pairs are related to (1,3)?
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  7. #7
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    (1,2)R(2,1) since 1+2=3 and 2+1=3

    (1,3) therefore is it related to (3,1) and (2,2) since when you add them up they all equal ?
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  8. #8
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    Quote Originally Posted by cooltowns View Post
    (1,2)R(2,1) since 1+2=3 and 2+1=3

    (1,3) therefore is it related to (3,1) and (2,2) since when you add them up they all equal ?
    Correct! Now finish.
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  9. #9
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    ok i am not sure exactly on how to finish the question

    but am i correct in saying all these are the combination of relations ?
    (1,1) =2
    (1,2),(2,1)=3
    (1,3),(3,1),(2,2)=4
    (1,4),(4,1),(2,3),(3,2)=5
    (4,2),(2,4),(3,3)=6
    (4,3),(3,4)=7
    (4,4)=8

    I know that for something to expressed in an equivalence class the notat [] is used. however i am just not sure on how to apply it to my question.
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  10. #10
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    Quote Originally Posted by cooltowns View Post
    ok i am not sure exactly on how to finish the question

    but am i correct in saying all these are the combination of relations ?
    (1,1) =2
    (1,2),(2,1)=3
    (1,3),(3,1),(2,2)=4
    (1,4),(4,1),(2,3),(3,2)=5
    (4,2),(2,4),(3,3)=6
    (4,3),(3,4)=7
    (4,4)=8
    CORRECT!
    I know that for something to expressed in an equivalence class the notat [] is used. however i am just not sure on how to apply it to my question.
    As for the notation that depends on your instructor/textbook.
    Here are two [(1,3)]_{\mathcal{R}}=\mathcal{R}_{(1,3)}=\{(1,3),(3,1),(  2,2)\}
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  11. #11
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    thanks plato for all your help.
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