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

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Discussion Overview

The discussion revolves around the mathematical implications of the statements a = c and b = d, particularly focusing on the properties of equality and operations such as addition and multiplication. Participants explore the usefulness and definitions related to these constraints, as well as their implications in algebraic structures.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that if a = c and b = d, then a + b = c + d and ab = cd, but question the usefulness of these statements.
  • Others argue that these constraints do not provide useful information, suggesting they lead to trivial identities like 0 = 0 and 1 = 1.
  • One participant proposes that the statements might indicate closure under addition and multiplication, and questions the existence of an algebra where self-equality (a = a) is false.
  • Another participant emphasizes that equality is an equivalence relation, implying that a = a must hold true for all a, and expresses confusion about the relevance of closure to the discussion.
  • Some participants discuss the concept of cancellation in algebra, with one noting that the definition of equality allows for operations to be performed on both sides of an equation.
  • A reference is made to Leibniz's law or "substitution for functions" as a potential name for the property being discussed, along with a mention that addition and multiplication must be functions for the property to hold.

Areas of Agreement / Disagreement

Participants express differing views on the usefulness and implications of the statements a = c and b = d, with no consensus reached regarding their significance or naming conventions.

Contextual Notes

Some participants highlight that the discussion involves foundational concepts in logic and algebra, but there are unresolved questions about the implications of these properties in various mathematical contexts.

Square1
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if a = c and b = d, then a +b = c + d, and ab = cd

What do we call that? Danke.
 
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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.
 
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?
 
TGlad said:
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.
 
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.
 
it's usually just called cancellation or right cancellation if you're working iirc
 
  • #10
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.
 
  • #11
Thank you all!
 

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