derived set

The selected set or sets, their permutations and derived sets form the basic material with which the composer works.

A derived set may refer to:

A derived set can be generated by choosing appropriate transformations of any trichord except 0,3,6, the diminished triad.

A derived set can also be generated from any tetrachord that excludes the interval class 4, a major third, between any two elements.

Derived set (mathematics)

Calling a set S closed if will define a topology on the space in which is the derived set operator, that is, .

The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.

If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T space.

In addition to permutations, the basic row may have some set of notes derived from it, which is used to create a new row, these are derived sets.

Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology.

Hence derivative algebras stand to topological derived sets and WK4 as interior/closure algebras stand to topological interiors/closures and S4.

Later Cantor proved that if the set of zeros S is infinite, but the derived set S of S is finite, then the coefficients are all zero.

A derived set is one which is generated or derived from consistent operations on a subset, for example Webern's Concerto, Op.24, in which the last three subsets are derived from the first:

In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'.

Two subsets S and T are separated precisely when they are disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other).

Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derived set of the image of any subset is the image of the derived set of that subset.