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Set theoretical products

  1. Jul 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Let S be any set and T = ∅. What can you say about the set S×T?

    2. Relevant equations


    3. The attempt at a solution
    The solution is that S×T=∅. I'm not quite sure why this is though. Is it because there isn't anything in T to give an ordered pair so S×T is empty?
     
  2. jcsd
  3. Jul 29, 2016 #2

    PeroK

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    Can you find anything in ##S \times T##?
     
  4. Jul 29, 2016 #3

    Math_QED

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    Well, when we write out the definition of AxB, we have:

    AxB = {(a,b) | a ∈ A and b ∈ B}

    Now, apply this to your set, what do you notice?
     
  5. Jul 29, 2016 #4
    There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).
     
  6. Jul 29, 2016 #5

    PeroK

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    ##\emptyset \notin \emptyset##
     
  7. Jul 29, 2016 #6

    Math_QED

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    Let SxT = {(s,t)|s ∈ S and t ∈ T}
    Suppose that SxT ≠ ∅...

    Try to find a contradiction, then follows that S x T = ∅
     
  8. Jul 30, 2016 #7
    Ok thank you, that makes sense.
     
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