Definition 1.1.2: The Relation, R, in the example above is an Equivalence Relation


To demonstrate this we test it for the three properties necessary to make it one:


  1. Reflexivity
  2. Symmetry
  3. Transitivity


Test for reflexivity:


1R1, 2R2, 3R3, 4R4, 5R5, 6R6 all members of S satisfy this condition.


Test for symmetry:


In every case, the relation xRy is matched by yRx.


Test for Transitivity:


1R3 and 3R5 are satisfied by 1R3, and so on.




The Equivalence Relation R above can be viewed as establishing a relationship between all even numbers and all odd numbers in the set {1, 2, 3, 4, 5, 6}. We might say that 2 is equivalent to 4 in that they are both even numbers. Another way of saying this is that they are both divisible by 2 and leave a remainder of 0. We thus recognize that as numbers, 2 ¹ 4, but they do share a common property. (When we get to measurement we will need this feature. It is like the Mexican Sierra fish's description at the top of the Complexity Research Group's home page). Thus, an equivalence relation is a kind of generalization of the equality relation that allows us to focus on certain properties of our set. More accurately, numerical equality, 4 = 4, is a special equivalence relation among numbers, since it implies equivalence from any standpoint.