Proving Q is Transitive in Mendelson's Topology

[philosophking]

Hey everyone,

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.

 

[AKG]

I don't see why you're confused, everything is fine. Note that when you say, "a - c is in P and hence Q" what you really mean is that "a - c is in P and hence (a, c) is in Q".

 

[philosophking]

Oh right, thanks. I was confused because they gave a few unnecessary properties, and i really didn't understand it.

 

[Hurkyl]

This collection of properties is an important one -- the book presumably will either show or ask you to show other interesting properties that such a set P, and relation Q, would have.