Let D be the relation on Z defined as follows:
for all m,n in Z m D n <==> 3|(m^2 - n^2)
1.prove that D is an equivalence relation and
2.describe the distinct equivalence classes of D.
I don't know how to go abt it
To be an equivelence relation it must be
Reflexive yes becuause
symmetric yes because if
for some integer q, factoring we get
solving the 2nd for we get
sub this into the first one to get
This tells us that
for part two think graphically. Good luck.
I'll do part 1 . . .
Definition: .Let be the relation on defined as follows:
. . for all
1. Prove that is an equivalence relation.
. . . . . . . . (The difference of their squares is a multiple of 3.)
Reflexive: .Is ?
. . . . . Yes!
Symmetric: . ?
. . We have: .
. . Then: . . . . Yes!
Transitive: . ?
. . Add: . . . . Yes!
is reflexive, symmetric and transitive.
Therefore, is an equivalence relation.