# Equivalence classes

Question in attachment is as follows:

Consider all lines in the plane. If a relationship between 2 lines is defined by the expression that their slopes are equal, prove that this relationship is an equivalence relationship.

If we consider the set of all lines in the plane, how would you uniquely identify the equivalence classes?

(Formulas and remainder of question found in attachment)

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#### Solution Preview

In order to prove that a relationship is an equivalence relationship, one must show reflexivity, symmetry and transitivity.

Let x, y and z be lines in the plane

reflex: x R ...

#### Solution Summary

This shows how to uniquely identify the equivalence classes. The set of all lines in the planes are calculated.

$2.19