Is $\le$ defined by $\langle _1, _1 \rangle \le \langle _2, _2 \rangle$, if $_1 \le _2 \land_1 \ge _2$ a linear order?

Question

I need to tell if the relation $\le$ defined by relationship $\langle _1, _1 \rangle \le \langle _2, _2 \rangle$, if $_1 \le _2$ $\land$ $_1 \ge _2$ linear order?

I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!

Answer 1

A linear order needs two elements to be comparable i.e. for $\left<x_1,y_1\right>,\left<x_2,y_2\right>$ we have either $$\left<x_1,y_1\right>\preceq\left<x_2,y_2\right>$$ or $$\left<x_2,y_2\right>\preceq\left<x_1,y_1\right>.$$ If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.