My university "textbook" for discrete math is Schaum's Outline. In this outline he goes over Equivalence Relations and Partitions, and I got confused at a particular theorem.
From the book:
Theorem 2.6: Let R be an equivalence relation on a set S. Then S/R is a partition of S. Specifically:
(i) For each a in S, we have a ∈ [a].
(ii) [a] = [b] if and only if (a,b) ∈ R. (iii) If $[a] \ne [b]$, then [a] and [b] are disjoint. Conversely, given a partition {Ai} of the set S, there is an equivalence relation R on S such that the sets Aiare the equivalence classes. This important theorem will be proved in Problem 2.17.
EXAMPLE 2.13
(a) Consider the relation R = {(1,1),(1,2),(2,1),(2,2),(3,3)} on S = {1,2,3}. One can show that R is reflexive, symmetric, and transitive, that is, that R is an equivalence relation.
Also: [1] = {1,2},[2] = {1,2},[3] = {3} Observe that [1] = [2] and that S/R = {[1],[3]} is a partition of S. One can choose either {1,3} or {2,3} as a set of representatives of the equivalence classes.
My confusion arises from the S/R = {[1],[3]}. I don't understand how one can subtract a relation from a set of integers. What fundamental understanding am I missing?