Suppose U were the collection of all sets. Note that in particular U is a set, s

Question

Suppose U were the collection of all sets. Note that in particular U is a set, so we would have U U. This is not yet a contradiction; although most sets are not elements of themselves, perhaps some sets are elements of themselves. But it suggests that the sets in the universe U could be split into two categories: the unusual sets that, like U itself, are elements of themselves, and the more typical sets that are not. Let R be the set of sets in the second category. In other words, R = {A U | A A}. This means that for any set A in the universe U, A will be an element of R iff A A. In other words, we have A U(A R A A). Show that applying this last fact to the set R itself (in other words, plugging in R for A) leads to a contradiction. This contradiction was discovered by Bertrand Russell in 1901, and is known as Russell's Paradox.

Explanation / Answer

Answer:

Given that, in other words:

Since, R is a set of all sets then R belongs to the Universal set. Means, each set is a set to them self.

Given the sets in the sets does not belong to the elements/sets of themselves. If R is not an element/set of itself then it contradiction to the statement that set of all sets are not the sets of sets to themselves.

This is written in terms of the given statement as,

           

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