Sets

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Sets Before 19th century, sets were not considered to be mathematical objects. The modern study of set theory began with Georg Cantor. The theory developed during the early stage is generally called naive set theory. However, it was soon discovered that naive set theory leads to paradoxes. To resolve these paradoxes, mathematicians proposed axiomatic set theories, the core of which is the postulation of rules by which new sets can be
formed from given ones.

In this chapter, we will go through some axioms of set theory, but before the axioms, we discuss some basic operations on sets while assuming a set is well-defined. The definition of a set relies on the axioms, but we can discuss the operations first based on our naive understanding of sets. After that, our selection of axioms will be guided by the principle that they agree with our intuition as much as possible.

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