(adsbygoogle = window.adsbygoogle || []).push({}); Partial Order/Upper Bound Proof from "How to Prove It"

1. The problem statement, all variables and given/known data

Suppose R is a partial order on A and B is a subset of A. Let U be the set of all upper bounds for B. Prove that U is closed upward; that is, prove that if x E U and xRy then y E U.

2. Relevant equations

N/A

3. The attempt at a solution

This is Velleman's:

Suppose that x E U and xRy. To prove that y E U, we must show that y is an upper bound for B, so suppose that b E B. Since x E U, x is an upper bound for B,so bRx...

The proof goes on, but I am stuck at the bolded part. How do we know that the relation R is a relation such that, if x > b, then bRx? What if the relation R is defined as: {(x,y) E A x A | x >= y}? Isn't my definition of R reflexive, transitive, and antisymmetric, and therefore partially ordered, as the given states? And wouldn't my definition then mean that, if x > b, it is not true that bRx? Where am I going wrong?

Thanks in advance.

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# Homework Help: Partial Order/Upper Bound Proof from How to Prove It

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