Im unsure whether I have the correct answer here, can someone please verify
consider the relation $R$ on $\mathbb{Z}$ x $\mathbb{Z}$ defined by $(a,b)R(x,y)$ iff $a \leq x$ and $b \leq y$.
explain why this is not a total order relation:
a total order relation is partial order and has the property that any pair of elements $(a,b),(x,y)\in \mathbb{Z}$x$\mathbb{Z}$ are related to each other in the form $(a,b),(x,y)\in R$ or $(x,y),(a,b)\in R$
this relation is not total order because there exists a pair of elements in $\mathbb{Z}$ x $\mathbb{Z}$ eg. $(-1,-2),(-3,-4)$, this pair of elements cannot be related because $-1 \nleq -3$ and $-2 \nleq -4$ and the relation is anti symmetric which doesnt allow $(-3,-4),(-1,-2)$ to occur