I'm trying to prove $a\sim b \text{ if } a^2-b^2\geq0$. All examples I have been given have not included inequalities. For reflexive we can take $a^2-a^2\geq0$ for $x\in \mathbb{Z}$. What procedure do we use to see if it's symmetric?
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Are you talking about the relation $R$ given by $aRb\iff a^2-b^2 \geq 0$? Then for symmetry, try out a few examples. Do we have $1R2$? Do we have $2R1$? See what happens, and then write a proof based on that. – Arthur Oct 31 '17 at 10:26
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Is this the original task? This is not true. Take a=0 and b=1, then $0-1=-1<0$ – Cornman Oct 31 '17 at 10:27