Page 216 - HOW TO PROVE IT: A Structured Approach, Second Edition

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202 Relations
(a) Prove that U is closed upward; that is, prove that if x ∈ U and xRy,
then y ∈ U.
(b) Prove that every element of B is a lower bound for U.
(c) Prove that if x is the greatest lower bound of U, then x is the least upper
bound of B.
22. Suppose that R is a partial order on A, B 1 ⊆ A, B 2 ⊆ A, x 1 is the least
upper bound of B 1 , and x 2 is the least upper bound of B 2 . Prove that if
B 1 ⊆ B 2 then x 1 Rx 2 .
23. Prove Theorem 4.4.11.

4.5. Closures

According to the definition we gave in the last section, the relation L =
{(x, y) ∈ R × R | x ≤ y} is a total order on R, but the relation M ={(x, y) ∈
R × R | x < y} is not because it is not reflexive. Of course, these relations are
closely related. It’s clear that M ⊆ L, and the only ordered pairs in L that are
not in M are pairs of the form (x, x), for some x ∈ R. Note that all of these
ordered pairs must be in any reflexive relation on R. Thus, you could think of
L as being formed by starting with M and then adding just those ordered pairs
that must be added to create a reflexive relation. It follows that L is the smallest
relation on R that is reflexive and contains M as a subset. We are using the
word smallest here in exactly the way we defined it in the last section. If we let
F ={T ⊆ R × R | M ⊆ T and T is reflexive}, then L is the smallest element
of F, where as usual it is understood that we mean smallest in the sense of the
subset partial order. In other words, L is an element of F, and it’s a subset of
every element of F. We will say that L is the reflexive closure of M.

Definition 4.5.1. Suppose R is a relation on a set A. Then the reflexive closure
of R is the smallest set S ⊆ A × A such that R ⊆ S and S is reflexive, if there
is such a smallest set. In other words, a relation S ⊆ A × A is the reflexive
closure of R if it has the following three properties:

1. R ⊆ S.
2. S is reflexive.
3. For every relation T ⊆ A × A,if R ⊆ T and T is reflexive, then S ⊆ T .

According to Theorem 4.3.6, if a set has a smallest element, then it can
have only one smallest element. Thus, if a relation R hasareflexive closure,
then this reflexive closure must be unique, so it makes sense to call it the
reflexive closure of R rather than a reflexive closure. However, as we saw in