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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 primary criticisms of Forte's nomenclature are: (1) Forte's labels are arbitrary and difficult to memorize, and it is in practice often easier simply to list an element of the set class; (2) Forte's system assumes equal temperament and cannot easily be extended to include diatonic sets, pitch sets (as opposed to pitch-class sets), multisets ...
Quine's New Foundations (NF) set theory, in Quine's original presentations of it, treats the symbol = for equality or identity as shorthand either for "if a set contains the left side of the equals sign as a member, then it also contains the right side of the equals sign as a member" (as defined in 1937), or for "an object is an element of the set on the left side of the equals sign if, and ...
The set of integers and the set of even integers have the same order type, because the mapping is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite ), there ...
The set of the equivalence classes is sometimes called the quotient set or the quotient space of by , and is denoted by /. When the set S {\displaystyle S} has some structure (such as a group operation or a topology ) and the equivalence relation ∼ {\displaystyle \,\sim \,} is compatible with this structure, the quotient set often inherits a ...
In set theory, the kernel of a function (or equivalence kernel [1]) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", [2] or; the corresponding partition of the domain.
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.
In set theory, the intersection of two sets and , denoted by , [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to . [2] Notation and terminology