TITLE: why are equivalence relations called so?
QUESTION [6 upvotes]: "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?

REPLY [4 votes]: 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".