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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 parentp, 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: RealRangea&comma;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&equals;prop1..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.

See Also

assume

assume[parametric]

define

RealRange

 


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