Is relation xRy iff (x < 0) or (x ≥ 0 and x = y) reflexive, symmetric, transitive?

[Learning_Math]

1. R on the set (the reals) defined by xRy iff (x < 0) or (x > or equal to 0 and x = y)



2. None



3.

Reflexive - Yes, since no matter what x I choose, x will always be equal to x, and will therefore fit the conditions of the relation.

Symmetric - It is symmetric because if I choose a positive y, then it is symmetric because of (x > or equal to 0 and x = y). If I choose a negative y, it is symmetric because of (x < 0).

Transitive - I am a little unclear on this one. I think however, that it is transitive because if I choose a positive z, then it is transitive because of (y > or equal to 0 and y = z). If I choose a negative z, it is transitive because of (y < 0).

 

[grief]

I don't think it's symmetric. For example, it seems xRy but not yRx when x=-1 and y=1.

And I think you're right that it is transitive. However, it seems like you'd want to condition on x rather than z. That is, suppose xRy and yRz.
Then either x<0, in which case xRz. Or x>=0, in which case x=y so xRz because yRz.

 

[Learning_Math]

That was helpful, thank you grief.