Motivation: Significance of Equivalence Relations
The equivalence relation is one of the most important concepts in mathematics. This is because it has some unique and interesting properties. For instance, by the use of an equivalence relation $R\subset V\times V$ we can decompose the set into disjoint subsets of \(V\), called its equivalence classes or partitions. Moreover, every element of a class represents this class in a special way – if we can prove a mathematical theorem for this one element, the theorem is also proven for the whole class of elements. This is what makes equivalence classes particularly useful for mathematicians – the save much work.
Intuitively, we use equivalence classes from the very beginning of our life. For instance, each word we learn in our language is, in fact, an equivalence class. As an example, by saying the word “car”, our mind intuitively decomposes all other things into distinct equivalence classes, in which things which we say are “cars” form only one equivalence class. By saying “this car is red”, our mind intuitively decomposes this class into even more detailed equivalence classes of colored cars. Thus, all “red cars”, all “black cars”, and any other color of cars form distinct equivalence classes of the set of all “cars” in our mind.
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