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If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...
The two equivalent definitions are as follows. Using union and intersection: define [1] [2] = and = If these two sets are equal, then the set-theoretic limit of the sequence exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well.
"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought" [2]), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction.
This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it: [6] "One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion.
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...