Maximal Element need not be Unique

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Theorem

Let $\struct {S, \preccurlyeq}$ be an ordered set.

It is possible for $S$ to have more than one maximal element.


Proof

Consider the set $T$ defined as:

$T = \set {0, 1}$

Let $S$ be defined as:

$S := \powerset T \setminus T$

where $\powerset T$ denotes the power set of $T$.

That is:

$S = \set {\O, \set 0, \set 1}$


Let $\preccurlyeq$ be the relation defined on $S$ as:

$\forall a, b \in S: a \preccurlyeq b \iff a \subseteq b$

That is, $\preccurlyeq$ is the subset relation on $S$.

From Subset Relation is Ordering, $\struct {S, \preccurlyeq}$ is an ordered set.


Let $a \in S$ such that $\set 1 \preccurlyeq a$.

Then by inspection it is apparent that:

$a = \set 1$

That is, $\set 1$ is a maximal element of $\struct {S, \preccurlyeq}$.


Similarly, let $a \in S$ such that $\set 0 \preccurlyeq a$.

Then by inspection it is apparent that:

$a = \set 0$

That is, $\set 0$ is also a maximal element of $\struct {S, \preccurlyeq}$.


Hence $S$ has more than one maximal element.

$\blacksquare$


Also see


Sources