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Question on equivalence relation.

  1. Feb 14, 2007 #1
    let R be a reflexive and symmetric relation on a set X. let's define a relation S on X. by (x,y) in S iff there exists a finite sequence x0,x1,...,x_n
    of terms from X such that x0=x xn=y and (x_i,x_i+1) in R for i=0,1,...n-1.
    now ive proven that S is an equivalence relation, i need to show that S is the smallest equivalence relation that includes R, i.e if R is a subset of T then S is a subset of T.
    i think that the naswer is simple but perhaps too simple to be true.
    if (x,y) in S, then there exists a finite sequence x,y such that (x,y) in R and so (x,y) in T.
    am i right here?

    thanks in advance.
    i hope you can answer also my other threads which i have opened here, cause im having an exam in friday in this material and your help is understandably important for me.
     
  2. jcsd
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