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A graduate course notes on set theory, covering the basics, the axioms, the paradoxes, and the open problems. Learn about the language, the natural numbers, the ordinal numbers, the real numbers, and the universe of sets.
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- Sets
- Topological Spaces
- Proposition 9.
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set is a collection of objects, called its elements. We write x 2 A to mean that x is an element of a set A, we also say that x belongs to A or that x is in If A and B are sets, we say that B is a subset of A if every element of B is an element of A. In this case we also say that A. Two sets are considered equal i A B and B A. contains B, and we wr...
Let X be a set. A topology on X is a collection of subsets of X, ie,
Let (X; d) be a metric space. Then X is Hausdor ogy. with the metric topol- Let (X; <) be a totally ordered set. Then X is Hausdor topology. with the order
A PDF file with notes on set theory, functions, maps, and cardinality. Covers basic definitions, examples, properties, and theorems with proofs and exercises.
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Basic Set Theory. A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical in- duction, and formalizes the notion of mathematical functions. The material is mostly elementary. For those of you new to abstract mathematics elementary does not mean simple (though much of the ...
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Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe important properties of sets, and give examples.
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BABY SET THEORY We shall begin with an informal discussion of some basic concepts of set theory. In these days of the "new math," much of this material will be already familiar to you. Indeed, the practice of beginning each mathematics course with a discussion of set theory has become widespread, extending even to the elementary schools.
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A PDF document that introduces the primitive concepts of set theory and logic, such as belonging, designations, sentences, quantifiers, and axioms. It also explains the second De Morgan laws and gives examples of quantifier exchange.
Sets are fundamental building blocks of mathematics. While logic gives a language and rules for doing mathematics, set theory provides the material for building mathematical structures. Set theory is not the only possible framework. More recently one has used category theory as a foundation.
