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Let $\mathcal{R}$ denote a relation between vectors in $x, y \in \mathbb{R_+^2}$.

The relation is called complete if $\forall x, y \in \mathbb{R_+^2}$ we have $x\mathcal{R}y$ or $y\mathcal{R}x$.

The following relation is given $$(x_1,x_2)\mathcal{R}(y_1,y_2) \iff x_1\geq y_1 \text{ and } x_2\geq y_2.$$

If we take the vectors $x=(2,2)$ and $y = (3,1)$ then we have $\Big(\text{not } [x\mathcal{R}y]\Big)$ since $2 = x_1 < y_1 = 3 $ and also $\Big(\text{not } [y\mathcal{R}x]\Big)$ since $1 = y_2 < x_2=2 $.

Hence $\mathcal{R}$ is not a complete relation. Is this reasoning correct?

Keep_On_Cruising
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1 Answers1

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Its neither $x<y$ nor $x>y$. Thus the ordering relation is not total.

Wuestenfux
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