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What is the name of this?

  1. Oct 17, 2012 #1
    if a = c and b = d, then a +b = c + d, and ab = cd

    What do we call that? Danke.
     
  2. jcsd
  3. Oct 17, 2012 #2

    chiro

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    Hey Square1.

    I don't know any special name for those constraints: We just call it a set of constraints that tell us absolutely nothing useful.
     
  4. Oct 17, 2012 #3
    bah but they are useful!
     
  5. Oct 17, 2012 #4

    chiro

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    If a = c and b = d then a + b = c + d implies a + b = a + b which means 0 = 0. Also ab = cd implies ab = ab which implies 1 = 1 which again is useless.
     
  6. Oct 17, 2012 #5
    Maybe it is a statement that the algebra is closed under addition and multiplication, and all elements equal themselves (self-equality). Does anyone know of an algebra where a=a is false for some a?
     
  7. Oct 17, 2012 #6
    Equality is an equivalence relation; it is necessarily true that a = a for all a; and I can't really see what the statement would have to do with closure.
     
  8. Oct 17, 2012 #7
    Well this is what allows you do claim "what you do to one side, do to the other".

    I think the usefulness of it lays in the "usefullness" (sorry lol) of being able to write a = 5 on one side, and on the other side of an equation a = c = *something that has a very different looking form from 5*, for example an nasty integral, and make quick easy simplifications.

    This has piqued my interest because, replacing = with a congruence shows that the property is true in congruence equations. Addition and multiplication is defined in that system.

    I guess the real question is, if an operation is defined for a given system, must the operation follow the "what you do to one side must be done to the other side" rule to maintain the relation.
     
  9. Oct 18, 2012 #8
    No, the definition of equality allows you to do that.
     
  10. Oct 18, 2012 #9
    it's usually just called cancellation or right cancellation if you're working iirc
     
  11. Oct 18, 2012 #10

    micromass

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    This is a question about logic. Here are the axioms of equality: http://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms

    Apparently, it is known as Leibniz law or "substitution for functions".

    Also, in order for the thing to work, it is crucial that + and . are functions. So you could justify the property by saying that + and . are functions.
     
  12. Oct 18, 2012 #11
    Thank you all!
     
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