Exercice :
On pose \( E= \left\{ a\ ; \ b \right\} \)
1) Déterminer \( \mathcal{P} (E) \)
2) Déterminer \( \mathcal{P} ( \mathcal{P} (E)) \)
Corrigé :
1) Déterminons \(\mathcal{P}(E)\) :
On a \(E=\{a,b\}\).
Les parties de \(E\) sont \(\varnothing\), \(\{a\}\), \(\{b\}\) et \(\{a,b\}\).
Donc\[\mathcal{P}(E)=\{\varnothing,\{a\},\{b\},\{a,b\}\}.\]
2) Déterminons \(\mathcal{P}(\mathcal{P}(E))\) :
Puisque \(\mathcal{P}(E)\) possède \(4\) éléments, l'ensemble \(\mathcal{P}(\mathcal{P}(E))\) possède \(2^4=16\) éléments.
On obtient :\[\begin{aligned}\mathcal{P}(\mathcal{P}(E))=\{&\varnothing,\\&\{\varnothing\},\{\{a\}\},\{\{b\}\},\{\{a,b\}\},\\&\{\varnothing,\{a\}\},\{\varnothing,\{b\}\},\\&\{\varnothing,\{a,b\}\},\{\{a\},\{b\}\},\\&\{\{a\},\{a,b\}\},\{\{b\},\{a,b\}\},\\&\{\varnothing,\{a\},\{b\}\},\\&\{\varnothing,\{a\},\{a,b\}\},\\&\{\varnothing,\{b\},\{a,b\}\},\\&\{\{a\},\{b\},\{a,b\}\},\\&\{\varnothing,\{a\},\{b\},\{a,b\}\}\}.\end{aligned}\]