Let m be a positive integer and define a relation R on the...

Let $m$ be a positive integer and define a relation $R$ on the set $X$ of all nonnegative integers by a $R b$ if and only if $a$ and $b$ have the same remainder when divided by $m$. Prove that $R$ is an equivalence relation on $X$. How many different equivalence classes does this equivalence relation have?

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Key Concepts

Equivalence Relation

An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. This means every element is related to itself (reflexivity), if one element is related to another then that second element is related to the first (symmetry), and if one element is related to a second and the second is related to a third then the first is related to the third (transitivity). Establishing these properties shows that the relation partitions the set into distinct classes.

Congruence Modulo m

Congruence modulo m is a relation between integers where two numbers are considered equivalent if they have the same remainder when divided by a positive integer m. This concept is fundamental in modular arithmetic and shows how numbers can be grouped into classes based on their remainder, forming a natural example of an equivalence relation.

Equivalence Classes and Partitioning

Equivalence classes are the subsets formed when an equivalence relation is applied to a set, with each class containing elements that are equivalent to each other. In the case of congruence modulo m, these classes correspond to the different possible remainders (from 0 to m-1), thereby partitioning the set of integers into m distinct equivalence classes.

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