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omoplata
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If [itex]\mathbb{Z}[/itex] is the set of integers, what does [itex]\mathbb{Z/2Z}[/itex] mean?
HallsofIvy said:Note that although henry_m and Micromass are using different words, they are saying the same thing!
Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It is a fundamental theory that forms the basis for other mathematical theories and has applications in various fields such as computer science, physics, and statistics.
The basic concepts of Set theory include sets, elements, subsets, unions, intersections, and functions. Sets are collections of objects, while elements are the individual objects within a set. Subsets are sets that contain only elements from a larger set. Unions and intersections are operations that combine or compare sets, respectively. Functions are mathematical relationships between sets.
Set theory is considered the foundation of mathematics as it provides a rigorous way of defining mathematical objects and operations. It is also closely related to other branches of mathematics such as logic, algebra, and topology.
The axioms of Set theory are a set of fundamental assumptions that form the basis of the theory. They include the axiom of extension, which states that two sets are equal if they have the same elements, and the axiom of regularity, which ensures that there are no infinite descending chains of sets.
Set theory has many practical applications, such as in computer science, where it is used to model data structures and algorithms. It is also used in statistics to represent data sets and in physics to describe the behavior of systems with multiple components.