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The Order Axioms

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Use the field and order axioms to prove the following proposition:
If xy > 0, then either x > 0 and y > 0, or x < 0 and y < 0.

Please explain stepwise.

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Solution Preview

The Order Axioms are listed here, for example: http://math.wikia.com/wiki/Real_number#Order_axioms

We will use the property of the field:
For any x, we have (-1)x=-x, the opposite of x.
Proof: x+(-1)x = 1x+(-1)x = (1-1)x = 0x = 0

Suppose, xy>0.
By the Totality Axiom, either
a) x=0, but then xy = 0y = 0, ...

Solution Summary

The Order Axioms are depicted.