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Public-key Cryptography: Theory and Practice by C. E. Veni Madhavan, Abhijit Das

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2.2. Sets, Relations and Functions

Sets are absolutely basic entities used throughout the present-day study of mathematics. Unfortunately, however, we cannot define sets. Loosely speaking, a set is an (unordered) collection of objects. But we run into difficulty with this definition for collections that are too big. Of course, infinite sets like the set of all integers or real numbers are not too big. However, a collection of all sets is too big to be called a set. (Also see Exercise 2.6.) It is, therefore, customary to have an axiomatic definition of sets. That is to say, a collection qualifies to be a set if it satisfies certain axioms. We do not go into the details of this axiomatic definition, but tell the axioms as properties of sets. Luckily ...

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