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edited Jun 18, 2020 at 8:32

I am not sure if this is a valid answer to this question:

I would probably say that equivalence relation is a fancy name for a partition of sets and equivalence class is a fancy name for a cell of this partition. (A partition of a set $X$ is an expression for $X$ as a disjoint union of non-empty subsets of $X$.)

Indeed, one has the following "theorem":

Theorem. There is a one-to-one correspondence between equivalence relations on a set $X$ and partitions of the set $X$.

Proof. Note first that if $\sim$ is an equivalence relation on $X$, then, $\sim$-classes (non-empty by definition) partition $X$. Conversely, if $X = \sqcup_{\lambda \in \Lambda} X_\lambda$ is a partition of $X$, define a relation $\sim$ on $X$ by $x \sim y$ if $x, y \in X_\lambda$ for some $\lambda$. Note then that $\sim$ is an equivalence relation and $\sim$-classes are precisely the $X_\lambda$'s.

Then, this becomes a matter of language as it should.

I am not sure if this is a valid answer to this question:

I would probably say that equivalence relation is a fancy name for a partition of sets and equivalence class is a fancy name for a cell of this partition. (A partition of a set $X$ is an expression for $X$ as a disjoint union of non-empty subsets of $X$.)

Indeed, one has the following "theorem":

Theorem. There is a one-to-one correspondence between equivalence relations on a set $X$ and partitions of the set $X$.

Proof. Note first that if $\sim$ is an equivalence relation on $X$, then, $\sim$-classes (non-empty by definition) partition $X$. Conversely, if $X = \sqcup_{\lambda \in \Lambda} X_\lambda$ is a partition of $X$, define a relation $\sim$ on $X$ by $x \sim y$ if $x, y \in X_\lambda$ for some $\lambda$. Note then that $\sim$ is an equivalence relation and $\sim$-classes are precisely the $X_\lambda$'s.

Then, this becomes a matter of language as it should.

I am not sure if this is a valid answer to this question:

I would probably say that equivalence relation is a fancy name for a partition of sets and equivalence class is a fancy name for a cell of this partition. (A partition of a set $X$ is an expression for $X$ as a disjoint union of non-empty subsets of $X$.)

Indeed, one has the following "theorem":

Theorem. There is a one-to-one correspondence between equivalence relations on a set $X$ and partitions of the set $X$.

Proof. Note first that if $\sim$ is an equivalence relation on $X$, then, $\sim$-classes (non-empty by definition) partition $X$. Conversely, if $X = \sqcup_{\lambda \in \Lambda} X_\lambda$ is a partition of $X$, define a relation $\sim$ on $X$ by $x \sim y$ if $x, y \in X_\lambda$ for some $\lambda$. Note then that $\sim$ is an equivalence relation and $\sim$-classes are precisely the $X_\lambda$'s.

Then, this becomes a matter of language as it should.

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created Apr 10, 2014 at 18:04

kan

I am not sure if this is a valid answer to this question:

I would probably say that equivalence relation is a fancy name for a partition of sets and equivalence class is a fancy name for a cell of this partition. (A partition of a set $X$ is an expression for $X$ as a disjoint union of non-empty subsets of $X$.)

Indeed, one has the following "theorem":

Theorem. There is a one-to-one correspondence between equivalence relations on a set $X$ and partitions of the set $X$.

Proof. Note first that if $\sim$ is an equivalence relation on $X$, then, $\sim$-classes (non-empty by definition) partition $X$. Conversely, if $X = \sqcup_{\lambda \in \Lambda} X_\lambda$ is a partition of $X$, define a relation $\sim$ on $X$ by $x \sim y$ if $x, y \in X_\lambda$ for some $\lambda$. Note then that $\sim$ is an equivalence relation and $\sim$-classes are precisely the $X_\lambda$'s.

Then, this becomes a matter of language as it should.