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Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)

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Sets are well-determined collections that are completely characterized by their elements. Thus, two sets are equal if and only if they have exactly the same elements. The basic relation in set theory is that of elementhood, or membership. We write a ∈ A a∈A to indicate that the object a a is an element, or a member, of the set A A . We also say that a a belongs to A A . Thus, a set A A is equal to a set B B if and only if for every a a , a ∈ A a∈A if and only if a ∈ B a∈B . In particular, there is only one set with no elements at all. This set is called, naturally, the empty set, and is represented by the symbol ∅ ∅ . We say that A A is a subset of B B , written A ⊆ B A⊆B , if every element of A A is an element of B B . Thus, A = B A=B if and only if A ⊆ B A⊆B and B ⊆ A B⊆A . Notice that ∅ ⊆ A ∅⊆A , for every set A A . Given sets A A and B B , one can perform some basic operations with them yielding the following sets: The set A ∪ B A∪B , called the u

--> Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy) Stanford Encyclopedia of Philosophy Menu Browse Table of Contents What's New Random Entry Chronological Archives About Editorial Information About the SEP Editorial Board How to Cite the SEP Special Characters Advanced Tools Contact Support SEP Support the SEP PDFs for SEP Friends Make a Donation SEPIA for Libraries Entry Navigation Back to Entry Entry Contents Entry Bibliography Academic Tools Friends PDF Preview Author and Citation Info Back to Top Supplement to Set Theory Basic Set Theory Sets are well-determined collec

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