Which properties does this relation satisfy?

Question

The relation is $(x_1,y_1)R(x_2,y_2)$ such that $(x_1 ≥ x_2) ∧ (y_1 ≥ y_2)$. I am asked to decide which properties satisfy (reflexivity, symmetry and transitivity ) but I cannot understand how the relation works here.

Answer 1

One has that $(x,y)R(a,b)$ if and only if $a\leq x$ and $b\leq y$. Clearly $x\leq x$ and $y\leq y$, thus $(x,y)R(x,y)$ and hence $R$ is reflexive. Notice that $(2,3)R(1,3)$ but not $(1,3)R(2,3)$, hence $R$ is not symmetric. Now try to determine whether $R$ is anti-symmetric and transitive yourself.