Set theory

Ideal (set theory)

In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal. More formally, given a set an ideal on is a nonempty subset of the powerset of such that: 1. * 2. * if and then and 3. * if then Some authors add a fourth condition that itself is not in ; ideals with this extra property are called proper ideals. Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set. The dual notion of an ideal is a filter. (Wikipedia).

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Introduction to sets || Set theory Overview - Part 2

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From playlist Set Theory

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Introduction to sets || Set theory Overview - Part 1

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From playlist Set Theory

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From playlist Set Theory

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From playlist Axiomatic Set Theory

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From playlist Set Theory by Mathoma

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From playlist Sets

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From playlist Set Theory

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From playlist Spring 2023 Online Kolchin Seminar in Differential Algebra

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From playlist Lecture series on modified traces in algebra and topology

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From playlist Global Noncommutative Geometry Seminar (Americas)

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From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]

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From playlist Global Noncommutative Geometry Seminar (Americas)

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From playlist Set Theory

Related pages

Ideal (order theory) | Lebesgue measure | Set theory | Quotient set | Finite set | Ideal (ring theory) | Empty set | Ordinal number | Filter (set theory) | Natural number | Set (mathematics) | Symmetric difference | Union (set theory) | Real number | Cartesian product | Boolean ring | Subset | Counting measure | Bijection | Equivalence relation | Stationary set | Measure space | Image (mathematics) | Cofinality | Boolean algebra (structure)