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zooxanthellae
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Partial Order/Upper Bound Proof from "How to Prove It"
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.
N/A
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.
Homework Statement
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.
Homework Equations
N/A
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.