Equivalence relations and addition

In summary, the question asks for a proof that if a equals a' then a+b is equivalent to a' + b. However, this statement is not necessarily true for all equivalence relations, as it depends on the specific relation being used.
  • #1
The1TL
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Homework Statement



prove that if a~a' then a+b ~ a' + b

Homework Equations





The Attempt at a Solution



I can prove that if a=a' then a+b = a' + b but how can I apply this to any equivalence relation
 
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  • #2
Your question makes no sense at all. An equivalence relation can be established on any set whatsoever- I could, for example, say that two automobiles are equivalent if and only if they were manufactured by the same company- so "a+ b" makes no sense in general.

Further, even if we assume that you are talking about numbers, whether it is true that a+ b= a'+ b', depends upon exactly what the equivalence relation is! It is NOT true for any equivalence relation on numbers. I can, for example, define a~ b if and only if |a|= |b|. I can then take a= 5, a'= -5, b= 4, b'= 4. It is NOT true that a+ b= 5+ 4= 9 is equal to a'+ b'= -5+ 4= -1.
 

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that describes a relationship between two elements where they are considered equivalent or equal in some way.

2. How do you determine if a relation is an equivalence relation?

A relation is an equivalence relation if it satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element is related to itself, symmetry means that if two elements are related, then they are also related in the opposite direction, and transitivity means that if two elements are related to a third element, then they are also related to each other.

3. How does addition relate to equivalence relations?

Addition is a binary operation that can be used to create equivalence relations. When two elements are added together and the result is equal, they are considered equivalent. This is known as the equivalence class of addition.

4. Can an equivalence relation be defined using other mathematical operations besides addition?

Yes, an equivalence relation can be defined using any binary operation as long as it satisfies the three properties mentioned earlier: reflexivity, symmetry, and transitivity.

5. How are equivalence relations useful in mathematics?

Equivalence relations are useful in many areas of mathematics, including algebra, geometry, and number theory. They allow us to define relationships between elements and classify them into equivalence classes, which can help us solve problems and prove theorems.

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