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Exercise:
mathcalU : the set of all sets. mathcalR : mathcalP | mathcalPin mathcalUquad textandquad mathcalP notin mathcalP or in words the set of all sets that do not contain themselves. i.e. mathcalU in mathcalU. So mathcalU notin mathcalR. bf To be shown: mathcalR in mathcalR iff mathcalR notin mathcalR

Solution:
Proof. If mathcalR in mathcalR then by definition of mathcalR we have mathcalR notin mathcalR. This shows mathcalR in mathcalR Longrightarrow mathcalR notin mathcalR If mathcalR notin mathcalR then by definition mathcalR does contain itself so mathcalR in mathcalR. This shows mathcalR in mathcalR Longleftarrow mathcalR notin mathcalR
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Exercise:
mathcalU : the set of all sets. mathcalR : mathcalP | mathcalPin mathcalUquad textandquad mathcalP notin mathcalP or in words the set of all sets that do not contain themselves. i.e. mathcalU in mathcalU. So mathcalU notin mathcalR. bf To be shown: mathcalR in mathcalR iff mathcalR notin mathcalR

Solution:
Proof. If mathcalR in mathcalR then by definition of mathcalR we have mathcalR notin mathcalR. This shows mathcalR in mathcalR Longrightarrow mathcalR notin mathcalR If mathcalR notin mathcalR then by definition mathcalR does contain itself so mathcalR in mathcalR. This shows mathcalR in mathcalR Longleftarrow mathcalR notin mathcalR
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