if a = c and b = d, then a +b = c + d, and ab = cd
What do we call that? Danke.
I don't know any special name for those constraints: We just call it a set of constraints that tell us absolutely nothing useful.
bah but they are useful!
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.
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?
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.
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.
No, the definition of equality allows you to do that.
it's usually just called cancellation or right cancellation if you're working iirc
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.
Thank you all!
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