# Equivalence relations and classes

Show that if R1 and R2 are equivalence relations on a set X, then R1 is a subset of R2 iff every R2-class is the union of R1 classes.

Attempt: I don't understand that if R2 has elements nothing to do with the elements of R1, how can an R2 class be a union of those elements belonging to an R1 class?

HallsofIvy
Homework Helper
I'm not sure what you mean by "R2 has elements nothing to do with the elements of R1". Both R1 and R2 are defined as equivalence relations on X so they both consist of ordered pairs of elements of X. Doesn't one "have something to do" with the other?

It is, of course, possible that the two relations are dijoint, that is, that they have no elements in common. A simple example of this is X= {0, 1, 2, 3}, R1 defined by "xR1y if and only if x= y= 1" so that R1 is the set {(1, 1)} and R2 defined by "xR2y if and only if x=y= 3" so that R2 iis the set {(3, 3)}.
But in this case the "if and only if" statement says if one is true then the other is also. In this example, both parts of the statement, "R1 is a subset of R2" and "every R2-class is the union of R1 classes" are both false and so the statement itself is true. The only way you could have a counter-example would be if one of the two statements was true and the other was false.

Actually I didn't even realize that the statements were false... But why is "R1 is a subset of R2" false? Could you explain this as if I were a 5 year old?

haruspex