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- Fréchet_inequalities abstract "In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George Boole and explicitly derived by Maurice Fréchet that govern the combination of probabilities about logical propositions or events logically linked together in conjunctions (AND operations) or disjunctions (OR operations) as in Boolean expressions or fault or event trees common in risk assessments, engineering design and artificial intelligence. These inequalities can be considered rules about how to bound calculations involving probabilities without assuming independence or, indeed, without making any dependence assumptions whatsoever. The Fréchet inequalities are closely related to the Boole–Bonferroni–Fréchet inequalities, and to Fréchet bounds.If Ai are logical propositions or events, the Fréchet inequalities areProbability of a logical conjunction (&)max(0, P(A1) + P(A2) + ... + P(An) − (n − 1)) ≤ P(A1 & A2 & ... & An) ≤ min(P(A1), P(A2), ..., P(An)),Probability of a logical disjunction (∨)max(P(A1), P(A2), ..., P(An)) ≤ P(A1 ∨ A2 ∨ ... ∨ An) ≤ min(1, P(A1) + P(A2) + ... + P(An)),where P( ) denotes the probability of an event or proposition. In the case where there are only two events, say A and B, the inequalities reduce toProbability of a logical conjunction (&)max(0, P(A) + P(B) − 1) ≤ P(A & B) ≤ min(P(A), P(B)),Probability of a logical disjunction (∨)max(P(A), P(B)) ≤ P(A ∨ B) ≤ min(1, P(A) + P(B)).The inequalities bound the probabilities of the two kinds of joint events given the probabilities of the individual events. For example, if A is "has lung cancer", and B is "has mesothelioma", then A & B is "has both lung cancer and mesothelioma", and A ∨ B is "has lung cancer or mesothelioma or both diseases", and the inequalities relate the risks of these events.Note that logical conjunctions are denoted in various ways in different fields, including AND, &, ∧ and graphical AND-gates. Logical disjunctions are likewise denoted in various ways, including OR, |, ∨, and graphical OR-gates. If events are taken to be sets rather than logical propositions, the set-theoretic versions of the Fréchet inequalities areProbability of an intersection of eventsmax(0,P(A) + P(B) − 1) ≤ P(A ∩ B) ≤ min(P(A), P(B)),Probability of a union of eventsmax(P(A), P(B)) ≤ P(A ∪ B) ≤ min(1, P(A) + P(B)).↑ ↑ ↑ ↑".
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- Fréchet_inequalities wikiPageWikiLink AND_gate.
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- Fréchet_inequalities wikiPageWikiLink Boolean_algebra.
- Fréchet_inequalities wikiPageWikiLink Booles_inequality.
- Fréchet_inequalities wikiPageWikiLink Category:Articles_containing_proofs.
- Fréchet_inequalities wikiPageWikiLink Category:Probabilistic_inequalities.
- Fréchet_inequalities wikiPageWikiLink Category:Probability_bounds_analysis.
- Fréchet_inequalities wikiPageWikiLink Category:Probability_theory.
- Fréchet_inequalities wikiPageWikiLink Category:Statistical_inequalities.
- Fréchet_inequalities wikiPageWikiLink Copula_(probability_theory).
- Fréchet_inequalities wikiPageWikiLink Engineering_design.
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- Fréchet_inequalities wikiPageWikiLink Event_trees.
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- Fréchet_inequalities wikiPageWikiLink Fréchet_bounds.
- Fréchet_inequalities wikiPageWikiLink Fréchet–Bell_inequalities.
- Fréchet_inequalities wikiPageWikiLink George_Boole.
- Fréchet_inequalities wikiPageWikiLink Independence.
- Fréchet_inequalities wikiPageWikiLink Independence_(probability_theory).
- Fréchet_inequalities wikiPageWikiLink Intersection_(set_theory).
- Fréchet_inequalities wikiPageWikiLink Logical_conjunction.
- Fréchet_inequalities wikiPageWikiLink Logical_disjunction.
- Fréchet_inequalities wikiPageWikiLink Logical_proposition.
- Fréchet_inequalities wikiPageWikiLink Maurice_René_Fréchet.
- Fréchet_inequalities wikiPageWikiLink Modus_ponens.
- Fréchet_inequalities wikiPageWikiLink OR_gate.
- Fréchet_inequalities wikiPageWikiLink Probabilistic_logic.
- Fréchet_inequalities wikiPageWikiLink Probability.
- Fréchet_inequalities wikiPageWikiLink Probability_bounds_analysis.
- Fréchet_inequalities wikiPageWikiLink Probability_distribution.
- Fréchet_inequalities wikiPageWikiLink Proposition.
- Fréchet_inequalities wikiPageWikiLink Reliability_theory.
- Fréchet_inequalities wikiPageWikiLink Risk_assessment.
- Fréchet_inequalities wikiPageWikiLink Risk_assessments.
- Fréchet_inequalities wikiPageWikiLink Set_(mathematics).
- Fréchet_inequalities wikiPageWikiLink Set_theory.
- Fréchet_inequalities wikiPageWikiLink Stochastic_dependence.
- Fréchet_inequalities wikiPageWikiLink Union_(set_theory).
- Fréchet_inequalities wikiPageWikiLinkText "Boole–Fréchet inequalities".
- Fréchet_inequalities wikiPageWikiLinkText "Boole–Fréchet inequality".
- Fréchet_inequalities wikiPageWikiLinkText "Fréchet inequalities".
- Fréchet_inequalities hasPhotoCollection Fréchet_inequalities.
- Fréchet_inequalities wikiPageUsesTemplate Template:Reflist.
- Fréchet_inequalities subject Category:Articles_containing_proofs.
- Fréchet_inequalities subject Category:Probabilistic_inequalities.
- Fréchet_inequalities subject Category:Probability_bounds_analysis.
- Fréchet_inequalities subject Category:Probability_theory.
- Fréchet_inequalities subject Category:Statistical_inequalities.
- Fréchet_inequalities comment "In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George Boole and explicitly derived by Maurice Fréchet that govern the combination of probabilities about logical propositions or events logically linked together in conjunctions (AND operations) or disjunctions (OR operations) as in Boolean expressions or fault or event trees common in risk assessments, engineering design and artificial intelligence.".
- Fréchet_inequalities label "Fréchet inequalities".
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- Fréchet_inequalities sameAs Q5506660.
- Fréchet_inequalities sameAs Q5506660.
- Fréchet_inequalities wasDerivedFrom Fréchet_inequalities?oldid=511173080.
- Fréchet_inequalities isPrimaryTopicOf Fréchet_inequalities.