What is meant by equivalence class?

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What is meant by equivalence class?
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When the elements of some set S have a notion of equivalence
(formalized as an equivalence relation) defined on them,
then one may naturally split the set S into equivalence classes.

These equivalence classes are constructed so that elements a and b belong to
the same equivalence class if, and only if, they are equivalent.

Ex.
1) Let X be the set of all rectangles in a plane,
and ~ the equivalence relation "has the same area as",
then for each positive real number A,
there will be an equivalence class of all the rectangles that have area A.

2) If X consists of all the lines in, say, the Euclidean plane,
and L ~ M means that L and M are parallel lines,
then the set of lines that are parallel to each other form an equivalence class,
as long as a line is considered parallel to itself.
In this situation, each equivalence class determines a point at infinity.