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Set Theory - An Introduction

Author: Sam Greeves
by Sam Greeves
Posted: May 05, 2014

Introduction

Set theory is a branch of mathematical logic, which focuses on the study of sets, which happen to be the collection of objects. Though any kind of object can be collected into a set, set theory is usually applied to concepts which are more relevant to maths. Set theory's language may be used in defining all types of mathematical objects. Modern study of set theory was started by George Cantor and Richard Dedekind in 1870s. After paradoxes were discovered in naive set theory, many axiom systems were put forward in early 20 th century by different mathematicians. Among all of them, Zermelo-Fraenkel axioms, along with axiom of choice is the most well known. The Zermelo-Fraenkel theory is often used as a foundation system for mathematics.

Basic concepts & notation

It starts with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o? A. As sets are objects, the membership relation can relate sets also. A derived binary relation in the middle of two sets is the subset relation, which is also called set inclusion. If all members of set A are members of set B as well, then A is a subset of B and will be denoted as A? B. If for instance, (1,2) is a subset of {1,2,3}, but {2,4} isn't. From this application it is clear that a set is its own subset. In cases where we try to remove this rule, proper subset is defined. A is a proper subset of B provided A is a subset of B, but B isn't a subset of A.

Applications

Many concepts of mathematics can be explained with precision only with the help of set theoretic concepts. For instance, different mathematical structures as vector spaces, manifolds, graphs and rings can be explained as sets which satisfy (axiomatic) properties. Equivalence and order relations are omnipresent in all branches of maths and mathematical theory relations can be explained in set theory. Since it was published for the first time in Principia Mathematica, scholars have claimed that all theories of mathematics can be explained using suitably designed sets of axioms of set theory with the help of first or second order logic. This theory is the basis for mathematical analysis, abstract algebra, topology as well as discrete maths.

Conclusion

Mathematicians have accepted that axioms of state theory can be used to describe theorems in different areas of maths such as topology, abstract algebra and discrete maths. Properties of real as well as natural numbers can be also derived with the help of set theory; since each number system can be detected using a set of equivalence relation whose field is infinite set. Set theory has many subdivisions such as combinational, fuzzy, inner model and descriptive set theory.

About the Author

This article has been compiled by sam g, who is an educational instructor.

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Author: Sam Greeves

Sam Greeves

Member since: Apr 21, 2014
Published articles: 19

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