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sparkster
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When I took a math foundations class we only did naive set theory and took as an axiom that the empty set is a member of every set. The book had formal set theory and thus listed the ZFC axioms. One of them was that the empty set exists and that it was a member of every set. I've looked at a couple of listing of the axioms on the net and they only give the existence of the empty set.
To prepare for graduate analysis (I start grad school in a week), I've been reading through Rudin's Principles of Mathematical Analysis. I noticed today that an exercise in the book is to prove that the empty set is a member of every set.
I'm not asking for the solution, but I was wondering how one would go about proving this. Whenever you prove subsets, you chase elements. But the empty set has no elements. Are there other methods of proving subsets?
To prepare for graduate analysis (I start grad school in a week), I've been reading through Rudin's Principles of Mathematical Analysis. I noticed today that an exercise in the book is to prove that the empty set is a member of every set.
I'm not asking for the solution, but I was wondering how one would go about proving this. Whenever you prove subsets, you chase elements. But the empty set has no elements. Are there other methods of proving subsets?