Let A={1,2,3,..,14,15}.Define the relation R on A*A (A squared) by(a,b)R(c,d) if a+d=b+c
find all the equivalence classes determined by R. represent the equivalence classes on the plane. what do the equivalence classes represent?
Well, it's easy to find the equivalence classes.
Start with, say, (1, 1).
For another point (x, y) to be equivalent to this we require that 1 + y = 1 + x. So solve this for y and you get the equation y = x. Thus all the points in A X A that are equivalent to (1, 1) are in the form (a, a), where $\displaystyle a \in A$.
Now pick the point (2, 1), etc.
-Dan