Mathematik-Online lexicon: Example of Relations

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	Mathematik-Online lexicon:
	Example of Relations

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The inclusion of sets $\subseteq$ is a partial order in the power set ${\cal P}(M)$ of a set

since it is

reflexive ( $A\subseteq A$ ),

asymmetric ( $A\subseteq B \land B \subseteq A \Rightarrow A=B$ ),

and

transitive ( $A\subseteq B\subseteq C\Rightarrow A\subseteq C$ ).

However, if contains more than one element, then the inclusion is not an order:

$\displaystyle a,b \in M, a\neq b: \{a\} \not\subseteq \{b\} \land \{b\} \not\subseteq \{a\} \,,$

i.e. it is not complete.

The relation ,,has an equal number of elements `` is an equivalence relation in the power set ${\cal P}(M)$ of a finite set since it is

reflexive ( $\vert A\vert=\vert A\vert$ ),

symmetric ( $\vert A\vert=\vert B\vert \Rightarrow \vert B\vert=\vert A\vert$ ),

and

transitive ( $\vert A\vert=\vert B\vert=\vert C\vert \Rightarrow \vert A\vert =\vert C\vert$ ).

(Authors: Hörner/Abele)

see also:

automatisch erstellt am 11. 6. 2007