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Two sets [twelve-tone series], P and P ′ will be considered equivalent [equal] if and only if, for any p i,j of the first set and p ′ i ′,j ′ of the second set, for all is and js [order numbers and pitch class numbers], if i=i ′, then j=j ′. (= denotes numeral equality in the ordinary sense).
The set of all equivalence classes in with respect to an equivalence relation is denoted as /, and is called modulo (or the quotient set of by ). [3] The surjective map x ↦ [ x ] {\displaystyle x\mapsto [x]} from X {\displaystyle X} onto X / R , {\displaystyle X/R,} which maps each element to its equivalence class, is called the canonical ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
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Let Set* denote the groupoid of sets and bijections. That is, the category whose objects are (all) sets, and morphisms are (all) bijections. Proposition. [7] Each echelon construction scheme leads to a functor from Set* to itself. In particular, the permutation group of a set X acts on every scale set S X.
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