# Comparisons

# Why

We want language and notation involving order.

# Comparisons

A comparison is a
statement (see Statements) involving a partial (which may or may not
be total) order.

## Notation

Let $A$ be a set.
We tend to denote an arbitrary partial order
on $A$ by $\preceq$.
So $(A, \preceq)$ is a partially ordered set.

As usual (see Relations), we write $a \preceq b$ to mean $(a, b)
\in A$.
Alternatively, we write $b \succeq a$ to mean
$a \preceq b$.
In other words, $\succeq$ is the inverse
relation (see Converse Relations) of $\preceq$.

## Predecessors and successors

If $a \preceq b$ and $a \neq b$, we write $a
\prec b$ and say that $a$
precedes $b$.
In this case we call $a$ the
predecessor of $b$.
Alternatively, under the same conditions, we
write $b \succ a$ and we say that $b$
succeeds $a$.
In this case we call $b$ the
successor of $a$.

## Induced partial orders

Of course, the object we have defined and
denoted by $\prec$ is a relation on $A$.
It satisfies (i) for no elements $x$ and $y$
do $x \prec y$ and $y \prec x$ hold
simultaneously and (ii) if $x \prec y$ and $y
\prec z$, then $x \prec z$ (i.e., $\prec$ is
transitive).
It is worthwhile to observe that if $S$ is a
relation satisfying (i) and (ii), then the
relation $R$ defined to mean $(a, b) \in S$ or
$a = b$ is a partial order on $A$.

## Strict and weak relations

This connection between $\preceq$ and $\prec$
can be generalized.
The strict relation
corresponding to a relation $R$ on a set $A$
is the relation $S$ on $A$ defined by $(a, b)
\in S$ if $(a, b) \in R$ and $a \neq b$.
The weak relation
corresponding to a relation $S'$ on a set
$A$ is the relation $R'$ defined by $(a, b)
\in R'$ if $(a, b) \in S'$ or $a = b$.
For this reason, a relation is said to
partially order a set if
it is a partial order or if its corresponding
weak relation is one.