in if . Determine if this is an equivalence relation.
I know that an equivalence relationship requires it to be reflexive, symmetric and transitive. I know how to do reflexive and symmetric, but I'm having some trouble with transitive. I think I may have gotten it, or maybe I'm close.
Here's my attempt:
First introduced as a third variable for the transitive test. I said that if and were true, then and .Since as proven in the first step by assuming , and likewise for , n, p and m are all not zero.And therefore and are both true. Referring to the definition of a transitive relation that says , I say that this IS a transitive relation because both of the required relations in the implication of the definition (replacing x with n, y with p and z with m) are true, and therefore xRz is true, or in my case, nRm, which it what I was out to prove.
But now that I look at it more closely... what if p was negative? Then this wouldn't be transitive. I think I'm assuming something I'm not supposed to assume.
I was looking at an example in my book, and they are proving that it's not transitive if it's simply nm >= 0. They said that if aRb and bRc, then ab>=0 and bc>=0. Thus acb^2 >= 0. What allowed them to combine those two expressions to get that b^2? If I know that, I think I can just say that (in my problem) p^2 must be > 0 because all numbers squared are positive except for zero, but I've already proved that it's not zero.
A push in the right direction would be perfect. Please don't just spit out the answer