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Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from theZFC axioms and using first order logic.

Q: Application of set theory in real life?

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Im not sure if there is any application of set notation and set theory, however set notation is important when you start learning about the domains and ranges of functions.

Set theory does have many practical business applications. Some of these include how the Turing machine is used in computer science, and the theory of crisis of foundations in math.

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When you are trying to find the unknown concentrations in equilibrium reaction ( chemistry ) the result if the ICE table set up devolves into a quadratic equation. Happens in physics to.

It develops the power to apply logic and logic is an intigral part of mathematics. More over application of venn diagrams helps to get solutions of complicated questions easily. Set theory is a combination of art, logic and calculations.

Classical theory is a reference to established theory. Fuzzy set theory is a reference to theories that are not widely accepted.

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Set theory was founded by Georg Cantor in 1873.

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Set theory is the mathematical study of sets. Set theory in business is important because it assists with the operations and planning in a business.