Equivalence Classes: N-cross-N w/ {a,b}R{j,k} means a+k=b+j

[mxchickmagnet86]

Just working on some more studying and I found a relation that I can't seem to think up the equivalence classes for:

the relation is defined as R on N x N and (a,b) R (j,k) means that a+k=b+j. R is also an equivalence relation.

 

[mxchickmagnet86]

Re: Equivalence Classes

am I on the right track with something like

[(0,0)] = {(a,b) in N and a=b : (x,y) in N and x=y}

is one of them

another is
[(0,1)] = (a,b) in N : (x,y) in N with x=b and y=a

?

 

[daon]

Re: Equivalence Classes

Just working on some more studying and I found a relation that I can't seem to think up the equivalence classes for. the relation is defined as R on N x N and (a,b) R (j,k) means that a+k=b+j. R is also an equivalence relation.

Two elements (x1,y1) and (x2,y2) are related if: x1+y2 = x2+y1

Rewrite as: x2-x1 = y2-y1, and assume (x1,y1) is not equal to (x2,y2).

Then, (y2-y1)/(x2-x1) = 1.

Does this form of the relation look familiar to you?