A Question on equivalence?

Question

With respect to geometric terms, what is a typical eqivalence class.
And how do the set of distinct equivalence classes partitions in R2.

Note : here we are talking about a relation ~ on R2 by
(x1, y1) ~(x2, y2) iff x1=x2

Answer 1

HOMEWORK!

The equivalence class of (x,y) will be the set of all (x1,y1) with (x,y)~(x1,y1). In other words
{(x1,y1):x=x1}.

This is a vertical line with at x. Notice that R^2 is partitioned into the collection of vertical lines.

Answer 2

An equivalence class is a set of elements which "look like the same" from a particular perspective. This may or may not have a geometric picture attached to it. In your example, like mathematician explained above, the classes are all the elements who share the same x coordinate, i.e. elements who lie on the same vertical line x=a for some real number a. Given ANY relation, the equivalence classes would partition IR^2 into disjoint sets.