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I'm working through the first chapter of Mendelson's Topology right now and ran into this question:

Let P be a subset of real numbers R such that i) 1 is in P, 2) if a,b are in P then a+b are in P, and 3) for each x in R, either x is in P, x=0, or -x is in P. Define Q= {(a,b) such that (a,b) is in R x R and a-b is in P}. Prove that Q is transitive.

The only reason I'm unsure about this is because my proof was very short and didn't involve 2 of the properties. This is what i said:

To prove Q is transitive, we prove that if aRb and bRc then aRc. Suppose aRb and bRc, then by definition of Q a-b is in P and b-c is in P (and hence in Q). According to property 2 then, (a-b)+(b-c) is in P, or a-c is in P and hence Q, so Q is transitive.

See why I'm confused? Did I miss something?

Thanks for your help.