Proving antisymmetry for this partial ordered set

Question

I need to prove the anti-symmetric property of the following relation (the set is the cross-product of all positive integers).

(x1,x2) R (y1,y2) <=> EITHER (x1 + x2 < y1 + y2) OR (x1 + x2 = y1 + y2 AND x1 =< y1)

As far as I can see, you will always end up with a contradiction, because when you "reverse" the relation (aRb bRa), either the first element will not satisfy x1 =< y1, and if they do, the sum of the tuples will no be equal.

A hint or something would be greatly appreciated

Answer 1

HINT: Suppose that $\langle x_1,x_2\rangle\,R\,\langle y_1,y_2\rangle$, but $\langle x_1,x_2\rangle\ne\langle y_1,y_2\rangle$. There are two possibilities:

1. $x_1+x_2<y_1+y_2$: Show that in this case it cannot be true that $\langle y_1,y_2\rangle\,R\,\langle x_1,x_2\rangle$.
2. $x_1+x_2=y_1+y_2$ and $x_1<y_1$: Again show that it cannot be true that $\langle y_1,y_2\rangle\,R\,\langle x_1,x_2\rangle$.

Conclude that if $\langle x_1,x_2\rangle\,R\,\langle y_1,y_2\rangle$ and $\langle y_1,y_2\rangle\,R\,\langle x_1,x_2\rangle$, then $\langle y_1,y_2\rangle=\langle x_1,x_2\rangle$.