If we have a set S with an Equivalence Relation on it, then an equivalence class is a subset of S containing all the elements that are equivalent to each other.
As an example, we might have cars for sale on a car lot, all with different VIN, but we decide to declare equivalence based on color. One subset includes all the red cars.
There are as many equivalence classes as there are elements in the set, since we can take any element, hunt for equivalent elements, and put them together to form an equivalence class.
Appendix A
We are going to construct an example with set C = {red1, red2, red3, blue1, blue2, blue3}
The “short version” of it is that if two named cars have the same color for the word, they are equivalent. R is the Equivalence Relation.
- red 1 R red1
- red1 R red2
- red2 R red1
- red1 R red3
- red3 R red1
- red2 R red3
- red3 R red2
- blue1 R blue2
- blue2 R blue1
- blue1 R blue3
- blue3 R blue1
- blue2 R blue3
- blue3 R blue2
We use the above for the formal definition given below:
For each y∈C, define the subset R[y] of C as follows:
R[y]={x∈C | x R y}.
The strategy is to find your chosen value for y on the right and then whatever is on the left is in the Equivalence Class.
We write out the six Equivalent Classes as follows:
- R[red1] = {red1, red2, red3}
- R[red2] = {red1, red2, red3}
- R[red3] = {red1, red2, red3}
- R[blue1] = {blue1, blue2, blue3}
- R[blue2] = {blue1, blue2, blue3}
- R[blue3] = {blue1, blue2, blue3}
You can see that for any two sets, either the two sets are equal (all elements are the same) or the two sets are disjoint (there are no elements in their intersection–they share no elements in common).
Rob Sterling 