# Section 3.2: Problem 3 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let
be a model of
(or equivalently a model of
). For
and
in
define the equivalence relation:
Let
be the equivalence class to which
belongs. Order equivalence classes by
Show that this is a well-defined ordering on the set of equivalence classes.

First, we argue that
is an equivalence relation. The reflectiveness and symmetry of the relation follows immediately from the definition. Regarding the transitivity, we can use the axioms, in particular, S2, to see that if
then
iff
, and
iff
, from which the transitivity of
follows.

Now, we show that
is well-defined. Suppose that
. We need to show that if
then
, and if
, then
. If
, then
(
is an equivalence relation), and either
or
, in which case
, or
, in which case if we assume
or
, then
for some
(by S2), contradicting the assumption that
. In either case,
and
, implying
. Similarly, we consider the case
to show
.

Finally, we ensure that the ordering properties hold for
.

- Trichotomy. For every and , either ( ) or , hence, , and either ( ) or ( ).
- Antisymmetry. For every and , if and ( ) then not ( ). Alternatively, or ( ) iff ( ) or (hence, ) and ( ).
- Transitivity. For every , and , if and ( ) and and ( ), then and ( ). Here, we have , as otherwise, , and , implying for some , , and (by L1).