Consider the following equivalence relation on the set :
Notice that the elements of are split up into three groups, with everything in one group related to everything else in the same group, and nothing in one group related to anything in any other group.
In this example, the three groups are, and .
In fact, this is a general phenomenon, and the groups often represent something important. For example, if , then the groups of related people would be called "families". Two people are in the same family if and only if they are related to each other. were a set of people and R was the is-related-to relation
In general, the groups are referred to as equivalence classes. Formally, we have the following definition:
In other words,
Note: I usually use the symbols [ and ] instead of ⟦ and ⟧, but the wiki syntax makes this difficult. You may use either notation.When is clear from context, we just write ⟦a⟧.
In the above example, we have three equivalence classes: , , and .
You can see that Claim:⟦a⟧=⟦b⟧ if and only if aRb.
One more piece of terminology: