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.)

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SOLUTION  

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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