may

05-29-2007, 12:19 PM

I don't really know how this work, but I have a question. In class, we've just gone over equivalence classes. The book doesn't really go over it too well. Anyways, the problem I'm stuck with is as follows:

Allow S = N x N, where N is the set of all natural numbers, and allow rho to equal a binary relation on S defined by (x,y) rho (z,w) <-> x +y = z +w. Show that rho is an equivalence relation on S and describe the resulting equivalence classes.

This is how I started. I showed reflexivity, (x,y) rho (x,y), because y = y here. I then showed symmetricity. If (x,y) rho (z,w), then y = w so w = y and (z, w) rho (x,w).

Next, I did transitivity. I said if (x,y) rho (z,w) and (z,w) rho (s,t), then y = w and w = t, so y = t and (x,y) rho (s,t).

I have no idea on how to incorporate the x + y = z +w, nor figure out what the equivalence classes are.

Please help me!

Allow S = N x N, where N is the set of all natural numbers, and allow rho to equal a binary relation on S defined by (x,y) rho (z,w) <-> x +y = z +w. Show that rho is an equivalence relation on S and describe the resulting equivalence classes.

This is how I started. I showed reflexivity, (x,y) rho (x,y), because y = y here. I then showed symmetricity. If (x,y) rho (z,w), then y = w so w = y and (z, w) rho (x,w).

Next, I did transitivity. I said if (x,y) rho (z,w) and (z,w) rho (s,t), then y = w and w = t, so y = t and (x,y) rho (s,t).

I have no idea on how to incorporate the x + y = z +w, nor figure out what the equivalence classes are.

Please help me!