properties - Maple Help

property

description of properties used by assume

Description

 A property can be:
 • (1) a property name, for example, assume(f, continuous) or assume(x, rational). Property names are grouped into five groups: Aliased Names, Numeral Properties, Matricial Properties, Functional Properties, and Other Properties.
 The following table defines the names that are aliased to a property.

 Alias Property Description realcons OrProp(AndProp(real,constant), real_infinity) negative RealRange(-infinity,Open(0)) a real < 0 nonnegative RealRange(0,infinity) a real >= 0 positive RealRange(Open(0),infinity) a real > 0 natural AndProp(integer, RealRange(1,infinity)) an integer > 0 posint AndProp(integer, RealRange(1,infinity)) an integer > 0 odd LinearProp(2,integer,1) an integer of the form 2*integer+1 even LinearProp(2,integer,0) an integer of the form 2*integer

 The following table shows the properties for numerals, their parent(s) in the inclusion lattice, that is, if an object has property p then it also has property $\mathrm{parent}\left(p\right)$, and a description when it is not obvious.

 Name Parent Description complex TopProp NumeralNonZero NonZero, complex GaussianInteger complex complex numbers where both the real and imaginary parts are integers real complex imaginary complex complex numbers with the real part equal to zero (includes 0) rational real irrational real, NumeralNonZero GaussianPrime GaussianInteger, Gaussian integers with no Gaussian NumeralNonZero integer factors x, with |x|>1 integer GaussianInteger, rational fraction rational, non-integer rational NumeralNonZero prime integer composite integer an integer that is neither a prime nor a unit (includes all integers <-1) RealRange(x,y) real

 The following table shows the properties for functionals.

 Name Parent Description mapping TopProp a function (but the name "function" is a type name in Maple) unary mapping a function that takes only one parameter binary mapping monotonic mapping a function that over the reals and where defined is non-decreasing (increasing) OddMap mapping a unary function f(x) = -f(-x) EvenMap mapping a unary function f(x) = f(-x) continuous mapping a function that is continuous for every real value, in every parameter Strictly- Monotonic monotonic a function that is strictly increasing (or decreasing) where defined over the reals operator mapping a function mapping functions to functions differentiable continuous a function that has a derivative for every possible real value commutative binary Infinitely- Differentiable differentiable a function that has a derivative of any order for every real value PolynomialMap Infinitely- Differentiable LinearMap PolynomialMap, StrictlyMonotonic ArithmeticOper binary the five arithmetic operators (+,-,*,/,^) addmul ArithmeticOper, commutative

 The following table shows the properties for matricials. (Notation taken from the CRC Handbook of Mathematical Sciences, 5th edition)

 Name Parent antisymmetric SquareMatrix diagonal Hermitian, tridiagonal, LowerTriangular, UpperTriangular ElementaryMatrix SquareMatrix Hermitian symmetric idempotent SquareMatrix IdentityMatrix PositiveDefinite, ScalarMatrix, idempotent, NonSingular, antisymmetric LowerTriangular triangular matrix TopProp nilpotent SquareMatrix NullMatrix ScalarMatrix, singular, idempotent, nilpotent, antisymmetric NullVector vector PositiveDefinite PositiveSemidefinite, NonZero PositiveSemidefinite SquareMatrix RectangularMatrix matrix scalar vector, RectangularMatrix ScalarMatrix diagonal singular SquareMatrix SquareMatrix matrix symmetric SquareMatrix triangular SquareMatrix tridiagonal SquareMatrix UpperTriangular triangular vector matrix

 The following table shows other properties.

 Name Parent Description BottomProp No object has this property TopProp Every possible object has this property NonZero TopProp MutuallyExclusive property type property constant TopProp property TopProp

 • (2) most types (this includes constant values, for example, 0)
 • (3) numerical ranges: RealRange$a,b$, RealRange(-infinity, b), and RealRange(a, infinity) (where a and b can be either numeric values or Open(x) where x is a numeric value). Open(x) indicates that the range is open, that is, the end point x is excluded.
 • (4) AndProp(a, b, ...) the "and" expression of properties a, b, ... (where a, b, ... are properties as defined above). This property describes objects that have all the properties a, b, ...
 You can use And as a synonym for AndProp.
 • (5) OrProp(a, b, ...) the "or" expression of properties a, b, ... (where a, b, ... are properties as defined above). This property describes objects that have at least one of the properties a, b, ...
 You can use Or as a synonym for OrProp.
 • (6) Non(a) the "not" of the property a (where a is a property as defined above). This property describes objects that do not have property a.
 You can use Not as a synonym for Non.
 • (7) LinearProp(a, b, c) where a and c are of type complex(numeric) (or are expressions that evaluate to complex(numeric) when evalf is applied) and b is a property.  This allows the system to express properties like the odd integers: LinearProp(2,integer,1) or the imaginary integers: LinearProp(I,integer,0)
 • (8) property ranges:  prop1 .. prop2 (where prop1 and prop2 are properties and prop1 is included in prop2.  This property means that the object has at least prop2 but not less than prop1.  For example, integer .. rational properly describes the integers/2.  If $A=\mathrm{prop1}..\mathrm{prop2}$ then all possible y in prop1 have property A, and all possible z in A have property prop2.
 • (9) A parametric property, of the form propname(arg1,...), where propname is the name of the parametric property and arg1, ... are the parameters of the property.  These properties are unevaluated function calls. The function property/included/propname(a,b) should be defined and should test the inclusion of property a in property b, where at least one of a or b is a propname.