Ask¶

Module for querying SymPy objects about assumptions.

class sympy.assumptions.ask.AssumptionKeys[source]¶

This class contains all the supported keys by ask.

algebraic¶

Algebraic number predicate.

Q.algebraic(x) is true iff x belongs to the set of algebraic numbers. x is algebraic if there is some polynomial in p(x)\in \mathbb\{Q\}[x] such that p(x) = 0.

Examples

>>> from sympy import ask, Q, sqrt, I, pi
>>> ask(Q.algebraic(sqrt(2)))
True
>>> ask(Q.algebraic(I))
True
>>> ask(Q.algebraic(pi))
False

References

R1: https://en.wikipedia.org/wiki/Algebraic_number

antihermitian¶

Antihermitian predicate.

Q.antihermitian(x) is true iff x belongs to the field of antihermitian operators, i.e., operators in the form x*I, where x is Hermitian.

References

R2: http://mathworld.wolfram.com/HermitianOperator.html

bounded¶: See documentation of Q.finite.

commutative¶

Commutative predicate.

ask(Q.commutative(x)) is true iff x commutes with any other object with respect to multiplication operation.

complex¶

Complex number predicate.

Q.complex(x) is true iff x belongs to the set of complex numbers. Note that every complex number is finite.

Examples

>>> from sympy import Q, Symbol, ask, I, oo
>>> x = Symbol('x')
>>> ask(Q.complex(0))
True
>>> ask(Q.complex(2 + 3*I))
True
>>> ask(Q.complex(oo))
False

References

R3: https://en.wikipedia.org/wiki/Complex_number

complex_elements¶

Complex elements matrix predicate.

Q.complex_elements(x) is true iff all the elements of x are complex numbers.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.complex(X[1, 2]), Q.complex_elements(X))
True
>>> ask(Q.complex_elements(X), Q.integer_elements(X))
True

composite¶

Composite number predicate.

ask(Q.composite(x)) is true iff x is a positive integer and has at least one positive divisor other than 1 and the number itself.

Examples

>>> from sympy import Q, ask
>>> ask(Q.composite(0))
False
>>> ask(Q.composite(1))
False
>>> ask(Q.composite(2))
False
>>> ask(Q.composite(20))
True

diagonal¶

Diagonal matrix predicate.

Q.diagonal(x) is true iff x is a diagonal matrix. A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

Examples

>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix
>>> X = MatrixSymbol('X', 2, 2)
>>> ask(Q.diagonal(ZeroMatrix(3, 3)))
True
>>> ask(Q.diagonal(X), Q.lower_triangular(X) &
...     Q.upper_triangular(X))
True

References

R4: https://en.wikipedia.org/wiki/Diagonal_matrix

even¶

Even number predicate.

ask(Q.even(x)) is true iff x belongs to the set of even integers.

Examples

>>> from sympy import Q, ask, pi
>>> ask(Q.even(0))
True
>>> ask(Q.even(2))
True
>>> ask(Q.even(3))
False
>>> ask(Q.even(pi))
False

extended_real¶

Extended real predicate.

Q.extended_real(x) is true iff x is a real number or \(\{-\infty, \infty\}\).

See documentation of Q.real for more information about related facts.

Examples

>>> from sympy import ask, Q, oo, I
>>> ask(Q.extended_real(1))
True
>>> ask(Q.extended_real(I))
False
>>> ask(Q.extended_real(oo))
True

finite¶

Finite predicate.

Q.finite(x) is true if x is neither an infinity nor a NaN. In other words, ask(Q.finite(x)) is true for all x having a bounded absolute value.

Examples

>>> from sympy import Q, ask, Symbol, S, oo, I
>>> x = Symbol('x')
>>> ask(Q.finite(S.NaN))
False
>>> ask(Q.finite(oo))
False
>>> ask(Q.finite(1))
True
>>> ask(Q.finite(2 + 3*I))
True

References

R5: https://en.wikipedia.org/wiki/Finite

fullrank¶

Fullrank matrix predicate.

Q.fullrank(x) is true iff x is a full rank matrix. A matrix is full rank if all rows and columns of the matrix are linearly independent. A square matrix is full rank iff its determinant is nonzero.

Examples

>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix, Identity
>>> X = MatrixSymbol('X', 2, 2)
>>> ask(Q.fullrank(X.T), Q.fullrank(X))
True
>>> ask(Q.fullrank(ZeroMatrix(3, 3)))
False
>>> ask(Q.fullrank(Identity(3)))
True

hermitian¶

Hermitian predicate.

ask(Q.hermitian(x)) is true iff x belongs to the set of Hermitian operators.

References

R6: http://mathworld.wolfram.com/HermitianOperator.html

imaginary¶

Imaginary number predicate.

Q.imaginary(x) is true iff x can be written as a real number multiplied by the imaginary unit I. Please note that 0 is not considered to be an imaginary number.

Examples

>>> from sympy import Q, ask, I
>>> ask(Q.imaginary(3*I))
True
>>> ask(Q.imaginary(2 + 3*I))
False
>>> ask(Q.imaginary(0))
False

References

R7: https://en.wikipedia.org/wiki/Imaginary_number

infinite¶

Infinite number predicate.

Q.infinite(x) is true iff the absolute value of x is infinity.

infinitesimal¶: See documentation of Q.zero.

infinity¶: See documentation of Q.infinite.

integer¶

Integer predicate.

Q.integer(x) is true iff x belongs to the set of integer numbers.

Examples

>>> from sympy import Q, ask, S
>>> ask(Q.integer(5))
True
>>> ask(Q.integer(S(1)/2))
False

References

R8: https://en.wikipedia.org/wiki/Integer

integer_elements¶

Integer elements matrix predicate.

Q.integer_elements(x) is true iff all the elements of x are integers.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.integer(X[1, 2]), Q.integer_elements(X))
True

invertible¶

Invertible matrix predicate.

Q.invertible(x) is true iff x is an invertible matrix. A square matrix is called invertible only if its determinant is 0.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('Y', 2, 3)
>>> Z = MatrixSymbol('Z', 2, 2)
>>> ask(Q.invertible(X*Y), Q.invertible(X))
False
>>> ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z))
True
>>> ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
True

References

R9: https://en.wikipedia.org/wiki/Invertible_matrix

irrational¶

Irrational number predicate.

Q.irrational(x) is true iff x is any real number that cannot be expressed as a ratio of integers.

Examples

>>> from sympy import ask, Q, pi, S, I
>>> ask(Q.irrational(0))
False
>>> ask(Q.irrational(S(1)/2))
False
>>> ask(Q.irrational(pi))
True
>>> ask(Q.irrational(I))
False

References

R10: https://en.wikipedia.org/wiki/Irrational_number

is_true¶

Generic predicate.

ask(Q.is_true(x)) is true iff x is true. This only makes sense if x is a predicate.

Examples

>>> from sympy import ask, Q, symbols
>>> x = symbols('x')
>>> ask(Q.is_true(True))
True

lower_triangular¶

Lower triangular matrix predicate.

A matrix M is called lower triangular matrix if \(a_{ij}=0\) for \(i>j\).

Examples

>>> from sympy import Q, ask, ZeroMatrix, Identity
>>> ask(Q.lower_triangular(Identity(3)))
True
>>> ask(Q.lower_triangular(ZeroMatrix(3, 3)))
True

References

R11: http://mathworld.wolfram.com/LowerTriangularMatrix.html

negative¶

Negative number predicate.

Q.negative(x) is true iff x is a real number and \(x < 0\), that is, it is in the interval \((-\infty, 0)\). Note in particular that negative infinity is not negative.

A few important facts about negative numbers:

Note that Q.nonnegative and ~Q.negative are not the same thing. ~Q.negative(x) simply means that x is not negative, whereas Q.nonnegative(x) means that x is real and not negative, i.e., Q.nonnegative(x) is logically equivalent to Q.zero(x) | Q.positive(x). So for example, ~Q.negative(I) is true, whereas Q.nonnegative(I) is false.
See the documentation of Q.real for more information about related facts.

Examples

>>> from sympy import Q, ask, symbols, I
>>> x = symbols('x')
>>> ask(Q.negative(x), Q.real(x) & ~Q.positive(x) & ~Q.zero(x))
True
>>> ask(Q.negative(-1))
True
>>> ask(Q.nonnegative(I))
False
>>> ask(~Q.negative(I))
True

nonnegative¶

Nonnegative real number predicate.

ask(Q.nonnegative(x)) is true iff x belongs to the set of positive numbers including zero.

Note that Q.nonnegative and ~Q.negative are not the same thing. ~Q.negative(x) simply means that x is not negative, whereas Q.nonnegative(x) means that x is real and not negative, i.e., Q.nonnegative(x) is logically equivalent to Q.zero(x) | Q.positive(x). So for example, ~Q.negative(I) is true, whereas Q.nonnegative(I) is false.

Examples

>>> from sympy import Q, ask, I
>>> ask(Q.nonnegative(1))
True
>>> ask(Q.nonnegative(0))
True
>>> ask(Q.nonnegative(-1))
False
>>> ask(Q.nonnegative(I))
False
>>> ask(Q.nonnegative(-I))
False

nonpositive¶

Nonpositive real number predicate.

ask(Q.nonpositive(x)) is true iff x belongs to the set of negative numbers including zero.

Note that Q.nonpositive and ~Q.positive are not the same thing. ~Q.positive(x) simply means that x is not positive, whereas Q.nonpositive(x) means that x is real and not positive, i.e., Q.nonpositive(x) is logically equivalent to \(Q.negative(x) | Q.zero(x)\). So for example, ~Q.positive(I) is true, whereas Q.nonpositive(I) is false.

Examples

>>> from sympy import Q, ask, I
>>> ask(Q.nonpositive(-1))
True
>>> ask(Q.nonpositive(0))
True
>>> ask(Q.nonpositive(1))
False
>>> ask(Q.nonpositive(I))
False
>>> ask(Q.nonpositive(-I))
False

nonzero¶

Nonzero real number predicate.

ask(Q.nonzero(x)) is true iff x is real and x is not zero. Note in particular that Q.nonzero(x) is false if x is not real. Use ~Q.zero(x) if you want the negation of being zero without any real assumptions.

A few important facts about nonzero numbers:

Q.nonzero is logically equivalent to Q.positive | Q.negative.
See the documentation of Q.real for more information about related facts.

Examples

>>> from sympy import Q, ask, symbols, I, oo
>>> x = symbols('x')
>>> print(ask(Q.nonzero(x), ~Q.zero(x)))
None
>>> ask(Q.nonzero(x), Q.positive(x))
True
>>> ask(Q.nonzero(x), Q.zero(x))
False
>>> ask(Q.nonzero(0))
False
>>> ask(Q.nonzero(I))
False
>>> ask(~Q.zero(I))
True
>>> ask(Q.nonzero(oo))  #doctest: +SKIP
False

normal¶

Normal matrix predicate.

A matrix is normal if it commutes with its conjugate transpose.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.normal(X), Q.unitary(X))
True

References

R12: https://en.wikipedia.org/wiki/Normal_matrix

odd¶

Odd number predicate.

ask(Q.odd(x)) is true iff x belongs to the set of odd numbers.

Examples

>>> from sympy import Q, ask, pi
>>> ask(Q.odd(0))
False
>>> ask(Q.odd(2))
False
>>> ask(Q.odd(3))
True
>>> ask(Q.odd(pi))
False

orthogonal¶

Orthogonal matrix predicate.

Q.orthogonal(x) is true iff x is an orthogonal matrix. A square matrix M is an orthogonal matrix if it satisfies M^TM = MM^T = I where M^T is the transpose matrix of M and I is an identity matrix. Note that an orthogonal matrix is necessarily invertible.

Examples

>>> from sympy import Q, ask, MatrixSymbol, Identity
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('Y', 2, 3)
>>> Z = MatrixSymbol('Z', 2, 2)
>>> ask(Q.orthogonal(Y))
False
>>> ask(Q.orthogonal(X*Z*X), Q.orthogonal(X) & Q.orthogonal(Z))
True
>>> ask(Q.orthogonal(Identity(3)))
True
>>> ask(Q.invertible(X), Q.orthogonal(X))
True

References

R13: https://en.wikipedia.org/wiki/Orthogonal_matrix

positive¶

Positive real number predicate.

Q.positive(x) is true iff x is real and \(x > 0\), that is if x is in the interval \((0, \infty)\). In particular, infinity is not positive.

A few important facts about positive numbers:

Note that Q.nonpositive and ~Q.positive are not the same thing. ~Q.positive(x) simply means that x is not positive, whereas Q.nonpositive(x) means that x is real and not positive, i.e., Q.nonpositive(x) is logically equivalent to \(Q.negative(x) | Q.zero(x)\). So for example, ~Q.positive(I) is true, whereas Q.nonpositive(I) is false.
See the documentation of Q.real for more information about related facts.

Examples

>>> from sympy import Q, ask, symbols, I
>>> x = symbols('x')
>>> ask(Q.positive(x), Q.real(x) & ~Q.negative(x) & ~Q.zero(x))
True
>>> ask(Q.positive(1))
True
>>> ask(Q.nonpositive(I))
False
>>> ask(~Q.positive(I))
True

positive_definite¶

Positive definite matrix predicate.

If M is a :math:n \times n symmetric real matrix, it is said to be positive definite if \(Z^TMZ\) is positive for every non-zero column vector Z of n real numbers.

Examples

>>> from sympy import Q, ask, MatrixSymbol, Identity
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('Y', 2, 3)
>>> Z = MatrixSymbol('Z', 2, 2)
>>> ask(Q.positive_definite(Y))
False
>>> ask(Q.positive_definite(Identity(3)))
True
>>> ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
...     Q.positive_definite(Z))
True

References

R14: https://en.wikipedia.org/wiki/Positive-definite_matrix

prime¶

Prime number predicate.

ask(Q.prime(x)) is true iff x is a natural number greater than 1 that has no positive divisors other than 1 and the number itself.

Examples

>>> from sympy import Q, ask
>>> ask(Q.prime(0))
False
>>> ask(Q.prime(1))
False
>>> ask(Q.prime(2))
True
>>> ask(Q.prime(20))
False
>>> ask(Q.prime(-3))
False

rational¶

Rational number predicate.

Q.rational(x) is true iff x belongs to the set of rational numbers.

Examples

>>> from sympy import ask, Q, pi, S
>>> ask(Q.rational(0))
True
>>> ask(Q.rational(S(1)/2))
True
>>> ask(Q.rational(pi))
False

References

https://en.wikipedia.org/wiki/Rational_number

real¶

Real number predicate.

Q.real(x) is true iff x is a real number, i.e., it is in the interval \((-\infty, \infty)\). Note that, in particular the infinities are not real. Use Q.extended_real if you want to consider those as well.

A few important facts about reals:

Every real number is positive, negative, or zero. Furthermore, because these sets are pairwise disjoint, each real number is exactly one of those three.
Every real number is also complex.
Every real number is finite.
Every real number is either rational or irrational.
Every real number is either algebraic or transcendental.
The facts Q.negative, Q.zero, Q.positive, Q.nonnegative, Q.nonpositive, Q.nonzero, Q.integer, Q.rational, and Q.irrational all imply Q.real, as do all facts that imply those facts.
The facts Q.algebraic, and Q.transcendental do not imply Q.real; they imply Q.complex. An algebraic or transcendental number may or may not be real.
The “non” facts (i.e., Q.nonnegative, Q.nonzero, Q.nonpositive and Q.noninteger) are not equivalent to not the fact, but rather, not the fact and Q.real. For example, Q.nonnegative means ~Q.negative & Q.real. So for example, I is not nonnegative, nonzero, or nonpositive.

Examples

>>> from sympy import Q, ask, symbols
>>> x = symbols('x')
>>> ask(Q.real(x), Q.positive(x))
True
>>> ask(Q.real(0))
True

References

R15: https://en.wikipedia.org/wiki/Real_number

real_elements¶

Real elements matrix predicate.

Q.real_elements(x) is true iff all the elements of x are real numbers.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.real(X[1, 2]), Q.real_elements(X))
True

singular¶

Singular matrix predicate.

A matrix is singular iff the value of its determinant is 0.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.singular(X), Q.invertible(X))
False
>>> ask(Q.singular(X), ~Q.invertible(X))
True

References

R16: http://mathworld.wolfram.com/SingularMatrix.html

square¶

Square matrix predicate.

Q.square(x) is true iff x is a square matrix. A square matrix is a matrix with the same number of rows and columns.

Examples

>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix, Identity
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('X', 2, 3)
>>> ask(Q.square(X))
True
>>> ask(Q.square(Y))
False
>>> ask(Q.square(ZeroMatrix(3, 3)))
True
>>> ask(Q.square(Identity(3)))
True

References

R17: https://en.wikipedia.org/wiki/Square_matrix

symmetric¶

Symmetric matrix predicate.

Q.symmetric(x) is true iff x is a square matrix and is equal to its transpose. Every square diagonal matrix is a symmetric matrix.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('Y', 2, 3)
>>> Z = MatrixSymbol('Z', 2, 2)
>>> ask(Q.symmetric(X*Z), Q.symmetric(X) & Q.symmetric(Z))
True
>>> ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z))
True
>>> ask(Q.symmetric(Y))
False

References

R18: https://en.wikipedia.org/wiki/Symmetric_matrix

transcendental¶

Transcedental number predicate.

Q.transcendental(x) is true iff x belongs to the set of transcendental numbers. A transcendental number is a real or complex number that is not algebraic.

triangular¶

Triangular matrix predicate.

Q.triangular(X) is true if X is one that is either lower triangular or upper triangular.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.triangular(X), Q.upper_triangular(X))
True
>>> ask(Q.triangular(X), Q.lower_triangular(X))
True

References

R19: https://en.wikipedia.org/wiki/Triangular_matrix

unit_triangular¶

Unit triangular matrix predicate.

A unit triangular matrix is a triangular matrix with 1s on the diagonal.

Examples

>>> from sympy import Q, ask, MatrixSymbol
>>> X = MatrixSymbol('X', 4, 4)
>>> ask(Q.triangular(X), Q.unit_triangular(X))
True

unitary¶

Unitary matrix predicate.

Q.unitary(x) is true iff x is a unitary matrix. Unitary matrix is an analogue to orthogonal matrix. A square matrix M with complex elements is unitary if :math:M^TM = MM^T= I where :math:M^T is the conjugate transpose matrix of M.

Examples

>>> from sympy import Q, ask, MatrixSymbol, Identity
>>> X = MatrixSymbol('X', 2, 2)
>>> Y = MatrixSymbol('Y', 2, 3)
>>> Z = MatrixSymbol('Z', 2, 2)
>>> ask(Q.unitary(Y))
False
>>> ask(Q.unitary(X*Z*X), Q.unitary(X) & Q.unitary(Z))
True
>>> ask(Q.unitary(Identity(3)))
True

References

R20: https://en.wikipedia.org/wiki/Unitary_matrix

upper_triangular¶

Upper triangular matrix predicate.

A matrix M is called upper triangular matrix if \(M_{ij}=0\) for \(i<j\).

Examples

>>> from sympy import Q, ask, ZeroMatrix, Identity
>>> ask(Q.upper_triangular(Identity(3)))
True
>>> ask(Q.upper_triangular(ZeroMatrix(3, 3)))
True

References

R21: http://mathworld.wolfram.com/UpperTriangularMatrix.html

zero¶

Zero number predicate.

ask(Q.zero(x)) is true iff the value of x is zero.

Examples

>>> from sympy import ask, Q, oo, symbols
>>> x, y = symbols('x, y')
>>> ask(Q.zero(0))
True
>>> ask(Q.zero(1/oo))
True
>>> ask(Q.zero(0*oo))
False
>>> ask(Q.zero(1))
False
>>> ask(Q.zero(x*y), Q.zero(x) | Q.zero(y))
True

sympy.assumptions.ask.ask(proposition, assumptions=True, context={})[source]¶

Method for inferring properties about objects.

Syntax

ask(proposition)

ask(proposition, assumptions)

where proposition is any boolean expression

Examples

>>> from sympy import ask, Q, pi
>>> from sympy.abc import x, y
>>> ask(Q.rational(pi))
False
>>> ask(Q.even(x*y), Q.even(x) & Q.integer(y))
True
>>> ask(Q.prime(4*x), Q.integer(x))
False

Remarks

Relations in assumptions are not implemented (yet), so the following will not give a meaningful result.

>>> ask(Q.positive(x), Q.is_true(x > 0)) # doctest: +SKIP

It is however a work in progress.

sympy.assumptions.ask.ask_full_inference(proposition, assumptions, known_facts_cnf)[source]¶: Method for inferring properties about objects.

sympy.assumptions.ask.compute_known_facts(known_facts, known_facts_keys)[source]¶

Compute the various forms of knowledge compilation used by the assumptions system.

This function is typically applied to the results of the get_known_facts and get_known_facts_keys functions defined at the bottom of this file.

sympy.assumptions.ask.register_handler(key, handler)[source]¶

Register a handler in the ask system. key must be a string and handler a class inheriting from AskHandler:

>>> from sympy.assumptions import register_handler, ask, Q
>>> from sympy.assumptions.handlers import AskHandler
>>> class MersenneHandler(AskHandler):
...     # Mersenne numbers are in the form 2**n - 1, n integer
...     @staticmethod
...     def Integer(expr, assumptions):
...         from sympy import log
...         return ask(Q.integer(log(expr + 1, 2)))
>>> register_handler('mersenne', MersenneHandler)
>>> ask(Q.mersenne(7))
True

sympy.assumptions.ask.remove_handler(key, handler)[source]¶: Removes a handler from the ask system. Same syntax as register_handler

Ask¶

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