1.3 Exercise 1

An easy problem that introduces equivalence relations in somewhat a different scenario than the usual "clock arithmetic" ( \mathbb{Z} modulo  n ) context.

"Define two points  (x_0, y_0) and  (x_1, y_1) of the plane to be equivalent if  y_0 - x_0^2 = y_1 - x_1^2 .  Check that this is an equivalence relation and describe the equivalence classes."

(Taken from Topology by James R. Munkres, Second Edition, Prentice Hall, NJ, 2000. Page 28.)



Reflexivity, symmetry, and transitivity of the relation  C is but a re-statement of reflexive, symmetric, and transitive properties of equality.  

Reflexivity:  \forall x \in A, xCx means  \forall (x,y) \in \mathbb{R} \times \mathbb{R}, y + x^2 = y + x^2 \Rightarrow 0 = 0 is a statement of the identity property of equality.

Symmetry:  xCy \Rightarrow yCx means  y_0 - x_0^2 = y_1 - x_1^2 \Rightarrow y_1 - x_1^2 = y_0 - x_0^2 .  This is a statement of symmetry of equality.  

Transitivity:  xCy and  yCz \Rightarrow xCz means  y_0 - x_0^2 = y_1 - x_1^2   and  y_1 - x_1^2 = y_2 - x_2^2 \Rightarrow y_0 - x_0^2 = y_2 - x_2^2 , which is clear by transitivity of equality.

The family of standard parabolas  y = x^2 + K are the equivalence classes ( y - x^2 = K describes the parabolic level curves, and hence all points on a particular parabola are in a particular equivalence class).  The statement  y_0 - x_0^2 = y_1 - x_1^2 is a consequence of transitivity of equality on  K .

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