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Set theory is a branch of mathematical logic that deals with sets, which are collections of objects. It is the foundation of most of modern mathematics and provides a fundamental framework for building mathematical concepts and structures. The objects that make up a set are called elements or members of the set. Key concepts and topics in set theory include: Basic Definitions and Notation: Set theory starts with the basic concepts of sets and their members. Sets are usually denoted by capital letters, and elements are written inside curly braces. For example, A = {1, 2, 3} is a set containing the numbers 1, 2, and 3. Set Operations: There are several fundamental operations for combining sets, such as union (A ∪ B), intersection (A ∩ B), difference (A - B), and complement. Types of Sets: Set theory categorizes sets in various ways, such as finite and infinite sets, subsets, empty sets (null sets), and universal sets. Cardinality: This refers to the number of elements in a set. The concept is particularly important for comparing the sizes of infinite sets. Relations and Functions: Set theory provides the basis for defining mathematical relations and functions, which are sets of ordered pairs. Axiomatic Set Theory: Advanced set theory involves the study of sets under various axiomatic systems, the most common being Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Applications: Set theory is not only a foundational theory for mathematics but also has applications in computer science, logic, and probability theory. Set theory began with the work of Georg Cantor in the late 19th century and has since evolved into a major area of mathematical research with both practical and theoretical applications.
