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A simple problem on set theory

  1. Apr 20, 2013 #1


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    1. The problem statement, all variables and given/known data
    Let S be a set with an operation * which assigns an element a*b of S for any a,b in S. Let us assume that the following two rules hold:
    1. If a, b are any objects in S, then a*b = a
    2. If a, b are any objects in S, then a*b = b*a
    (Herstein, Abstract Algebra, 2ed)

    2. Relevant equations
    Is it safe to assume that the symmetry, transitivity, and reflexibility hold?

    3. The attempt at a solution
    a = a*b = b*a = b
    But I am not sure if this is sufficient as it is my first course (in fact, first problem!) in abstract algebra...
    Or with the relation obtained from the axioms of S, shall I proceed with proof by contradiction?
  2. jcsd
  3. Apr 20, 2013 #2


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    2017 Award

    Staff: Mentor

    This is true for all a,b in S, and shows that S just has one single element. That is a strange problem.
    The validity of those axioms will follow from that, but where is the point?
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