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Are the members of (x,y) if and only if (y = x + 3) & (y = x – 3) an empty set?

  1. Aug 24, 2012 #1
    Is the set of all integers where <x,y> [itex]\in[/itex] R if and only if (y = x + 3) & (y = x – 3) an empty set?
     
  2. jcsd
  3. Aug 24, 2012 #2

    chiro

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    Hey Bob4040 and welcome to the forums.

    Your definition is a little vague. You mention integers but then you say x and y are real numbers which is a bit of a contradiction.

    Are you trying to ask whether x and y are real numbers if the relation holds?
     
  4. Aug 24, 2012 #3
    I do not mean all real numbers. R is the set. x and y are members of the set on the condition that (y = x + 3) & (y = x – 3). These two lines never intersect, so I think that means it is an empty set?
     
  5. Aug 24, 2012 #4

    chiro

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    Ohh I see what you mean now.

    Well the easiest way to check is to set them equal to one another and see if you get a solution or a contradiction.

    y = x + 3, y = x - 3 implies x + 3 = x - 3 which implies 3 = -3 which is a contradiction, so you have proven that no element exists satisfying the relationship so you have the empty set.
     
  6. Aug 24, 2012 #5
    Thank you for the help!
     
  7. Aug 25, 2012 #6

    Bacle2

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    I know this is not what you meant/intended , Bob4040, but, strictly speaking your

    identity would hold if x were in the set called Z_3, where 3=-3 :

    http://www.wolframalpha.com/input/?i=Z+mod+3

    Then,for all x, x+3=x-3 .
     
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