. . . . . "16389"^^ . . . . . . . . . "In probabilistic logic, the Fr\u00E9chet inequalities, also known as the Boole\u2013Fr\u00E9chet inequalities, are rules implicit in the work of George Boole and explicitly derived by Maurice Fr\u00E9chet 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 Fr\u00E9chet inequalities are closely related to the Boole\u2013Bonferroni\u2013Fr\u00E9chet inequalities, and to Fr\u00E9chet bounds."@en . . . . "35736414"^^ . . . . . . "In probabilistic logic, the Fr\u00E9chet inequalities, also known as the Boole\u2013Fr\u00E9chet inequalities, are rules implicit in the work of George Boole and explicitly derived by Maurice Fr\u00E9chet 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 Fr\u00E9chet inequalities are closely related to the Boole\u2013Bonferroni\u2013Fr\u00E9chet inequalities, and to Fr\u00E9chet bounds. If Ai are logical propositions or events, the Fr\u00E9chet inequalities are \n* Probability of a logical conjunction \n* Probability of a logical disjunction 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 to \n* Probability of a logical conjunction \n* Probability of a logical disjunction 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 \u2228 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, &, \u2227 and graphical AND-gates. Logical disjunctions are likewise denoted in various ways, including OR, |, \u2228, and graphical OR-gates. If events are taken to be sets rather than logical propositions, the set-theoretic versions of the Fr\u00E9chet inequalities are \n* Probability of an intersection of events \n* Probability of a union of events"@en . . . . . . . . . . . . . . "Fr\u00E9chet inequalities"@en . . . . . . . . . . . . . . . . . . . . . . . . . . . "1100454928"^^ . . . . . . .