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