Consider the set $A=\{(x, y) \in \Bbb R\times\Bbb R \,|\, y \le 0\}.$
Prove that the following relation is of total order on $A.$ $$(x,y)\preceq(u,v)\iff (x<u) \vee [(x=u) \wedge (x^2+y^2 \ge u^2+v^2)]$$
2026-03-27 04:18:40.1774585120
Prove that a relation is a total order.
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