Every Set is an Element of its Power Set | Set Theory

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Every set is an element of its own power set. This is because the power set of a set S, P(S), contains all subsets of S. By definition, every set is a subset of itself, and thus by definition of the power set of S, it must contain S. This is even true for the always-fun empty set! We discuss these facts in this set theory video lesson.

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On one hand, every set A is a subset of A itself. On the other, the power set P(A) of a set A is the set containing all subsets of A. So A being a subset of A means by definition that A belongs to/is an element of P(A).

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