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Partition (mathematics)A partition of a set X is a set P of nonempty subsets of X such that every element x in X is in exactly one of these subsets. The elements of P are sometimes called the blocks of the partition.
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The set {1, 2, 3} has the following partitions
If an equivalence relation is given on the set X, then the set of all equivalence classes forms a partition of X. Conversely, if a partition P is given on X, we can define an equivalence relation on X by writing x ~ y iff there exists a member of P which contains both x and y. The notions of "equivalence relation" and "partition" are thus essentially equivalent.
The set of all partitions of a set is a partially ordered set; one may say that one partition is "finer" than another if it splits the set into smaller blocks. This partially ordered set is a lattice.
The Bell number Bn (named in honor of Eric Temple Bell) is the number of different partitions of a set with n elements. The first several Bell numbers are B0=1, B1=1, B2=2, B3=5, B4=15, B5=52, B6=203.
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banners and handkerchiefs, wept in excess of emotion. Bouquets of
every conceivable demonstration was made of the most enthusiastic
old grenadiers, whom he recognized, he called out of the ranks,
wants. He addressed several of the officers, whom he had seen in
pervaded all minds Upon his return to the Hotel deVille, he met
urging upon him the acceptance of the Presidency of the Cisalpine
and promised, on the following day, to meet the convention.
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of the occasion. Napoleon entered the church, took his seat upon an
and a large number of distinguished generals and statesmen. He
of manner, elegance of expression, and fluency of utterance as if
of oratory. He announced his acceptance of the dignity with which
which he adopted to secure the prosperity of the Italian Republic
applause interrupted his address, and at its close one continuous
the assembled multitude. Napoleon remained at Lyons twenty days,
then engrossed his attention. And yet he found time to write
government in France. The following brief extracts from this free
the motives which actuated Napoleon at this time, and of the great
mornings in giving audience to the deputations of the neighboring
During the past two years the population of Lyons has increased
works are in a state of high activity. All minds seem to be.
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