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Math Help - need help with equivalence class

  1. #1
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    need help with equivalence class

    For x belonging to Z, let us denote by [x] equivalence class which contains x. Prove that if [x]=[x'] and [y]=[y'] then [x+y]=[x'+y'] and [xy]=[x'y']
    Remark: When xRy people say that "x and y are congruent mod 3", and sometimes write x equivalent y (mod3)

    someone can help me please here???
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by tukilala View Post
    For x belonging to Z, let us denote by [x] equivalence class which contains x. Prove that if [x]=[x'] and [y]=[y'] then [x+y]=[x'+y'] and [xy]=[x'y']
    Remark: When xRy people say that "x and y are congruent mod 3", and sometimes write x equivalent y (mod3)

    someone can help me please here???
    If [x] denotes the equivalence class in \mathbb{Z} modulo 3 then all elements of [x] have the same remainder as x when divided by 3, and any integer which has this remainder is in [x].

    So:

    [x]=[x'] and [y]=[y'] means there are integers a,\ b,\ d,\ e,\ r_1,\ r_2 where r_1,\ r_2 \in \{0,1,2\} such that:

     <br />
x=3a+r_1,\ x'=3b+r_1,\ y=3d+r_2,\ y'=3e+r_2<br />


    Then x+y=3(a+d)+(r_1+r_2) , and so [x+y]=[r_1+r_2]

    Similarly:

    x'+y'=3(b+e)+(r_1+r_2) , and so [x'+y']=[r_1+r_2]

    Hence: [x+y]=[x'+y']

    CB
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