Are These Relations Symmetric?

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SUMMARY

The discussion focuses on determining the symmetry of five mathematical relations. The only relation confirmed to be symmetric is 2) x~y if and only if xy >= 0, as it holds true that if xy >= 0, then yx >= 0. Relation 5) x~y if and only if x+y is odd is also symmetric, since the sum of two numbers is invariant under order. Relations 1), 3), and 4) are not symmetric, with relation 3) being specifically questioned for its conditions.

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Determine which of these relations are symmetric

1) x~y if and only if x-y is positive
2) x~y if and only if xy >= 0
3) x~y if and only if x+2y is positive
4) x~y if and only if x+y is positive
5) x~y if and only if x+y is odd

I thought all but 1) but this was wrong.
The only one I am sure that is symmetric is 2)
 
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It is certainly true that if x+ y is positive, so is y+ x.

It is certainly true that if x+ y is odd, so is y+ x.

What about "x+ 2y is positive"? Can you find x such that x+ 2y is positive but y+ 2x is not?
 

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