Page 218 - HOW TO PROVE IT: A Structured Approach, Second Edition

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204 Relations
in which one object can be “strictly larger” than another. They are therefore
sometimes called strict partial orders.

Definition 4.5.3. Suppose R is a relation on A. Then R is said to be irreflexive
if ∀x ∈ A((x, x) /∈ R). R is called a strict partial order if it is irreflexive and
transitive. It is called a strict total order if it is a strict partial order, and in
addition it satisfies the following requirement, called trichotomy:

∀x ∈ A∀y ∈ A(xRy ∨ yRx ∨ x = y).

Note that the terminology here is slightly misleading. A strict partial order
isn’t a special kind of partial order. It’s not a partial order at all, since it’s not
reflexive! You may be surprised that we did not include antisymmetry in the
definition of strict partial order, since it was part of the definition of partial
order, but it turns out that antisymmetry is implied by the definition. For more
on this, see exercise 3.
You should be able to check for yourself that P is a strict partial order and
M is a strict total order. Perhaps you’ve already guessed from these examples
that the reflexive closure of a strict partial order is always a partial order, and
the reflexive closure of a strict total order is always a total order. You are asked
to prove this in exercise 4.
Reflexivity is not the only property for which we can define a closure. Exactly
the same idea could be applied to symmetry and transitivity.

Definition 4.5.4. Suppose R is a relation on A. The symmetric closure of R is
the smallest set S ⊆ A × A such that R ⊆ S and S is symmetric, if there is such
a smallest set. In other words, a relation S ⊆ A × A is the symmetric closure
of R if it has the following properties:

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

The transitive closure of R is the smallest set S ⊆ A × A such that R ⊆ S
and S is transitive, if there is such a smallest set. In other words, a relation
S ⊆ A × A is the transitive closure of R if it has the following properties:

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