why is an empty set a subset of itself

Note first of all the definition of a subset: We say A \subset B only when: if x \in A then x \in B.

The empty set is the set with no elements. It is denoted by the symbol \phi. A source of confusion to beginners is that although the empty set consists of nothing, it itself is something (namely, some particular set, the one characterized by the fact that nothing is in it). The set \{\phi \} is a set containing exactly one element, namely the empty set. (In a similar way, when dealing with numbers, say with ordinary integers, we must be careful not to regard the number zero as nothing; zero is something, a particular number, which represents the numbers in “nothing”. Thus, zero and \phi are quite different, but there is a connection between them in that the set has zero elements.) Note that for any set X, we have

\phi \subset X and X \subset X.

A special case of both of these statements is the statement  which occasions difficulty if, as is often improperly done, one reads “is contained in” for both of the symbols \phi and \in The statement

\phi \subset \phi

is true because the statement “for each x \in \phi, we have x \in \phi” is obviously true, and also because it is “vacuously true”, that is there is no for which the statement must be verified, just as the statement “all pigs with wings speak Chinese” is vacuously true.

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Do send your questions, suggestions, comments, more such seemingly paradoxical statements to me.

More later, happy new year 🙂

Nalin Pithwa

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