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Closed operation

  1. Mar 27, 2009 #1
    The question given is: Determine the smallest subset A of Z such that 2 ε A and A is closed with respect to addition.

    The answer given was the set of all positive even integers, but I was thinking that the smallest subset would be the given element and the identity element (0 in this case) so that A = {0,2}...wouldn't this be more accurate?
  2. jcsd
  3. Mar 27, 2009 #2
    To understand why that is not correct, you have to know what closure of an operator over a set means.

    A set S is closed under a binary operator + iff for all x, y in S, x + y is in S.

    A = {0, 2} isn't closed under addition because the definition is not satisfied. We can find a counter example where x and y are both in S, but x + y is not. The counter example is with x = 2 and y = 2.
  4. Mar 27, 2009 #3
    Closed means you can do it to any two (possibly non-unique) elements and get an answer in your set.

    S = {0, 2} doesn't work since 2+2=4 is not in S.
  5. Mar 27, 2009 #4
    I see. Thanks...funny I didn't notice that.
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