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# Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations.

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DEFINITIONS

A relation $R$ is an $\textbf{equivalence relation}$ if the relation $R$ is transitive, symmetric and reflexive.

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmetric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

The $\textbf{equivalence class}$ of $a$ is the set of all elements that are in relation to $a$.

Notation: $[a]_R$

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