What Mathematical Property Describes a = c and b = d?

[Square1]

if a = c and b = d, then a +b = c + d, and ab = cd

What do we call that? Danke.

 

[chiro]

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.

 

[Square1]

bah but they are useful!

 

[chiro]

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.

 

[TGlad]

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?

 

[Number Nine]

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.

 

[Square1]

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.

 

[Number Nine]

Well this is what allows you do claim "what you do to one side, do to the other".

No, the definition of equality allows you to do that.

 

[genericusrnme]

it's usually just called cancellation or right cancellation if you're working iirc

 

[micromass]

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.

 

[Square1]

Thank you all!