Is $\{((a,b),(c,d))|a+b\ge c+d\}$ an anti-symmetric relation?

Question

Question

Let $R$ be a relation on the set $\mathbb Z^+\times \mathbb Z^+$ defined as follows: $R=\{((a,b),(c,d))|a+b\ge c+d\}$.

Is this anti-symmetric?

Attempted Solution

I'd argue that it isn't. For $((a,b),(c,d))\in R$ and $((c,d),(a,b))\in R$ to be true, then $a+b$ must equal $c+d$. This is because -- per the definition above -- $a+b\ge c+d$. If $(a,b)>(c,d)$ then $((c,d),(a,b))$ could not exist in the relation, thus they must be equal in terms of sum. In such a scenario, we find that the above relation is not anti-symmetrical since $((10,5),(5,10))\in R$ and $((5,10),(10,5))\in R$ is true, but $(5,10)\ne (10,5)$.

Did I do this right?

Answer 1

I think it will be more clear if you take (1,3) and (2,2) as an example.