i have the following equivalence relation on R^2.
(x,y)~(x',y') iff there exists integers a,b such that x'=x+a+b and y'=y+b
i solved for y' to get y'=y+x'-x-a but im having trouble finding the equivalence classes. if someone could tell me the equivalence classes(either algebraically or geometrically)
will pay fg to anyone who can help. thanks

Your relation of equivalence can be described as :

(x;y) ~ (x';y') if and only if there exists integers a and b such that : x' = x + a , and y' = y + b.

Notice that "my" a corresponds to "your" a+b.

Geometrically : Let M (x;y) and M' (x';y') two points of the plane R².

M ~ M' if and only if vector(MM') has integer coordonates.

A point M being given, all the points M' in its equivalence class are reached with a Z² grid set to M.
You can describe the set of equivalence classes with the square [0;1[ x [0;1[ (with an opened border on the right, and on the top).