Every nite domain contains an identity element.

In summary, the conversation discusses a possible method for proving that every finite domain contains an identity element. The approach involves considering the operation in the domain as a mapping and manipulating numbers to show that an element can serve as an identity. However, it is uncertain if this method will work and further discussion is needed on how to approach the problem.
  • #1
Stephen88
61
0

Homework Statement


I'm trying to write a proof ot demonstrate that every finite domain contains
an identity element.

Homework Equations


The Attempt at a Solution


If I can think of the operation from the ring as a mapping...like x->yx..where y are just values from the domain and then to consider the possibility that for one y from the domain the following happens x=xy then maybe this will work.But not for addition
How should I think about this problem?
 
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  • #2
Oh wait but I can manipulate the numbers as I want...I can write x=y*d and y=y*e(this means that y doesn't have to be from the domain because I haven't proved the identity part) and =>x=y*e*d=>x=e*x=>e is and identity element in the domain.Will this work?
 

FAQ: Every nite domain contains an identity element.

What does "finite domain" mean?

A finite domain refers to a set of elements that has a limited or finite number of objects. In other words, it is a set with a definite and countable number of elements.

What is an identity element?

An identity element is an element within a set that, when combined with any other element using a specific operation, will produce the same element. In other words, the identity element is the element that does not change the result when used in an operation.

Why is it important for a finite domain to contain an identity element?

Having an identity element in a finite domain is important because it allows for the closure of operations. This means that when two elements in a finite domain are combined using an operation, the resulting element is also within the finite domain. This helps to ensure the consistency and completeness of mathematical systems.

Is the identity element unique in a finite domain?

Yes, the identity element is unique in a finite domain. This means that there can only be one identity element within a finite domain. If there were multiple identity elements, it would cause inconsistencies and contradictions in mathematical systems.

Can the identity element be different for different operations within the same finite domain?

No, the identity element must be the same for all operations within the same finite domain. This is necessary for the consistency and completeness of mathematical systems. If the identity element were different for different operations, it would cause contradictions and inconsistencies.

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