Equivalence classes have the following properties:In define a relation by:
~ if is divisible by .
Show that ~ is an equivalence relation.
1). Reflexivity a~a
2). Symmetry if a~b then b~a
3). Transitivity if a~b and b~c then a~c
1). Reflexivity: so reflexivity holds.
2). Symmetry: if then so symmetry holds.
3). Transitivity: if and then so transitivity holds.
I can see that this is going to have something to do with remainders, but i'm not really sure what. Surely remainders can be repeated?Let A denote the set of equivalence classes of ~ in .
Find an explicit bijection on .
Hint: What if the relation were " ~ if is divisible by "?