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Equivalence Relations

  1. Apr 14, 2010 #1
    1) Recall that an equivalence relation S on set R ( R being the reals) is a subset of R x R such that

    (a) For every x belonging to R (x,x) belongs to S
    (b) If (x,y) belongs to S, then (y,x) belongs to S
    (c) If (x,y) belongs to S and (y,z) belongs to S then (x,z) belongs to S

    What is the smallest equivalence relation S on the Set R of real numbers that contains all the points in the line y - x = 1. Prove your answer.

    Can anyone help figure this out? I am pretty lost on this one.
  2. jcsd
  3. Apr 14, 2010 #2


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    Have you tried drawing a picture of S? You know it has at least the line y - x = 1. What does (a) tell you in terms of your picture? And (b)? Does (c) work for your picture?
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