why are equivalence relations called so?

Question

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition.

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in X: a ~ a. (Reflexivity) if a ~ b then b ~ a. (Symmetry) if a ~ b and b ~ c then a ~ c. (Transitivity) "

But Why are "equivalence" relations called so?

Answer 1

The word "equivalent" is built from "equal" and "value" — well, actually from the Latin words aequus (equal) and valere (to be worth). So two elements are equivalent if they are in a sense equal-valued, or interchangeable. Often this is indeed used literally, by defining some quantities as equivalence classes; for example, the fraction $\frac42$ has the same value as (is equivalent to) the fraction $\frac21$.

Now the properties of an equivalence relation can be directly obtained from that interpretation of the word:

• Clearly some object has to have the same value as itself; thus any equivalence relation fulfils $a\sim a$ for all $a$ (reflexivity).
• Just as clearly, if $a$ has the same value as $b$, then $b$ has the same value as $a$. That is $a\sim b\implies b\sim a$ (symmetry).
• And of course, if $a$ has the same value as $b$, and $b$ has the same value as $c$, then $a$ has the same value as $c$, that is $a\sim b\land b\sim c\implies a\sim c$ (transitivity).

You may note that in the list above, I've ultimately used the exact same properties of equality. That is not accidental; equality is in a way the prototype of an equivalence relation.

Or in short, an equivalence relation describes a more general notion of "sameness".

Answer 2

I would guess it is because they partition the set into smaller sets, sometimes called equivalence classes. The idea is that the elements of each of these classes are all equivalent (when looked at in a specific way).

For example lets consider the relation $a\sim b$ if and only if $b-a$ is a multiple of $n$. This is an equivalence relation and it splits the integers into equivalence classes. In this case we say the elements of each class are all equivalent because they all give the same residue when we divide by $n$, so if we have an expression and only want to calculate the residue when dividing that expression by $n$ then we can swap some of the element of the expression for "equivalent" elements and this won't alter our result.