Determine which of the three properties (reflexive, symmetric, and transitive) apply to each relation. (a) Let S be the set of all lines in a plane and let R be the relation “is perpendicular to.” (b) Let S be the set of real numbers and let R be the relation “ >. ” (c) Let S be the set of all triangles in a plane and let R be the relation “is similar to.” Given an equivalence relation R on a set S, it is natural to group together all the elements that are related to a particular element. More precisely, we define the equivalence class (with respect to R ) of x ∈ S to be the set

 

 

Since R is reflexive, each element of S is in some equivalence class. Furthermore, two different equivalence classes must be disjoint. That is, if two equivalence classes overlap, they must be equal. To see this, suppose that w ∈ E_x ∩ E_y. Then for any x′ ∈ E_x we have x′R_x. But w ∈ E_x, so wR_x and, by symmetry, xR_w. Also, w ∈ E_y, so wR_y. Using transitivity twice, we have x′R_y, so that x′ ∈ E_y and E_x ⊆ E_y. The reverse inclusion follows in a similar manner.

 

 

Thus we see that an equivalence relation R on a set S breaks S into disjoint pieces in a natural way. These pieces are an example of a partition.