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In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other.
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more ...
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set . There are several variants and generalizations of the lexicographical ordering.
Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially ...
The order of flats is Bâ™, Eâ™, Aâ™, Dâ™, Gâ™, Câ™, and Fâ™ (reverse order of sharps), which can be remembered using the phrase: Battle Ends And Down Goes Charles' Father; Blanket Exploded And Dad Got Cold Feet. Before Eating A Doughnut Get Coffee First. To remember the difference between the whole rest and the half rest:
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [1] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
Mathematical structure. In mathematics, a structure is a set provided with some additional features on the set (e.g. an operation, relation, metric, or topology ). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possible structures are ...
Bag-of-words model. The bag-of-words model is a model of text which uses a representation of text that is based on an unordered collection (or "bag") of words. It is used in natural language processing and information retrieval (IR). It disregards word order (and thus any non-trivial notion of grammar [clarification needed]) but captures ...
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