The documentation for the computer language Java recommends that when an \"equal

Question

The documentation for the computer language Java recommends that when an "equals method" is defined for an object, it be an equivalence relation. That is, if R is defined as follows: xRy x. equals(y) for all objects in the class, then R should be an equivalence relation. Suppose that in trying to optimize some of the mathematics of a graphics application, a programmer creates an object called a point, consisting of two coordinates in the plane. The programmer defines an equals method as follows: if p and q are any points, then p. equals(q) the distance from p to q is less than or equal to c, where c is a small positive number that depends on the resolutions of the computer display. Is this programmer's equals method an equivalence relation? Justify your answer.

Explanation / Answer

equals method will be an equivalence method if it holds:
   reflexive, symmetric and transitive property.

The given programmer's equal methos is not equivalence relation

It does not hold transitive property:

   p1.equals(p2) => distance(p1, p2) <= c
   p2.equals(p3) => distance(p2, p3) <= c

   Then it is not ncecessary that : p1.equals(p3) => distance(p1, p3) <= c

   p1----(d1 <= c)------p2-------(d2 <= c)-------p3

   clearly (d1 + d2) can be greater than c

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