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Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
The notation / is also used, and is less ambiguous. Denotes the set of rational numbers (fractions of two integers). It is often denoted also by . Denotes the set of p-adic numbers, where p is a prime number.
In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1. The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).
The difference between the interval vector of a set and that of its complement is <X, X, X, X, X, X/2>, where (in base-ten) X = 12 – 2C, and C is the cardinality of the smaller set. In nearly all cases, complements of unsymmetrical sets are inversionally related—i.e. the complement of an "A" version of a set of cardinality C is (usually ...
The notation [,) is used to indicate an interval from a to c that is inclusive of —but exclusive of . That is, [ 5 , 12 ) {\displaystyle [5,12)} would be the set of all real numbers between 5 and 12, including 5 but not 12.
Successive Z-related hexachords from act 3 of Wozzeck [4]: 79 Play ⓘ. In musical set theory, a Z-relation, also called isomeric relation, is a relation between two pitch class sets in which the two sets have the same intervallic content (and thus the same interval vector) but they are not transpositionally related (are of different T n-type ) or inversionally related (are of different T n /T ...
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .