g$$ddlmZddlmZGddeZGddeZGddeZGd d eZGd d eZGd deZ GddeZ GddeZ GddeZ GddeZ GddeZGddeZy)) Predicate) Dispatcherc(eZdZdZdZeddZy)IntegerPredicatean Integer predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Integer integerIntegerHandlerzXHandler for Q.integer. Test that an expression belongs to the field of integer numbers.docN__name__ __module__ __qualname____doc__namerhandlerV/var/www/openai/venv/lib/python3.12/site-packages/sympy/assumptions/predicates/sets.pyrrs!0 DKGrrc(eZdZdZdZeddZy)NonIntegerPredicatez. Non-integer extended real predicate. nonintegerNonIntegerHandlerzYHandler for Q.noninteger. Test that an expression is a non-integer extended real number.r Nr rrrrr&s! DIGrrc(eZdZdZdZeddZy)RationalPredicatea Rational number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Rational_number rationalRationalHandlerzZHandler for Q.rational. Test that an expression belongs to the field of rational numbers.r Nr rrrrr2s!4 DLGrrc(eZdZdZdZeddZy)IrrationalPredicatea Irrational number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Irrational_number irrationalIrrationalHandlerzIHandler for Q.irrational. Test that an expression is irrational numbers.r Nr rrrrrUs 8 D9Grrc(eZdZdZdZeddZy) RealPredicatea Real number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Real_number real RealHandlerzRHandler for Q.real. Test that an expression belongs to the field of real numbers.r Nr rrrr"r"zs"8r DHGrr"c(eZdZdZdZeddZy)ExtendedRealPredicatea Extended real predicate. Explanation =========== ``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 extended_realExtendedRealHandlerzHandler for Q.extended_real. Test that an expression belongs to the field of extended real numbers, that is real numbers union {Infinity, -Infinity}.r Nr rrrr&r&s!0 DEGrr&c(eZdZdZdZeddZy)HermitianPredicatez Hermitian predicate. Explanation =========== ``ask(Q.hermitian(x))`` is true iff ``x`` belongs to the set of Hermitian operators. References ========== .. [1] https://mathworld.wolfram.com/HermitianOperator.html hermitianHermitianHandlerz^Handler for Q.hermitian. Test that an expression belongs to the field of Hermitian operators.r Nr rrrr*r*s! DOGrr*c(eZdZdZdZeddZy)ComplexPredicatea Complex number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Complex_number complexComplexHandlerzXHandler for Q.complex. Test that an expression belongs to the field of complex numbers.r Nr rrrr.r.s!6 DKGrr.c(eZdZdZdZeddZy)ImaginaryPredicatea Imaginary number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Imaginary_number imaginaryImaginaryHandlerzHandler for Q.imaginary. Test that an expression belongs to the field of imaginary numbers, that is, numbers in the form x*I, where x is real.r Nr rrrr2r2s 6 D=Grr2c(eZdZdZdZeddZy)AntihermitianPredicateaJ Antihermitian predicate. Explanation =========== ``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 ========== .. [1] https://mathworld.wolfram.com/HermitianOperator.html antihermitianAntiHermitianHandlerzHandler for Q.antihermitian. Test that an expression belongs to the field of anti-Hermitian operators, that is, operators in the form x*I, where x is Hermitian.r Nr rrrr6r6@s!" DOGrr6c(eZdZdZdZeddZy)AlgebraicPredicatea# Algebraic number predicate. Explanation =========== ``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 ========== .. [1] https://en.wikipedia.org/wiki/Algebraic_number algebraicAlgebraicHandlerzHandler for Q.algebraic key.r N)r r rrrrr<rrrr:r:[s6 D! .rr:c(eZdZdZdZeddZy)TranscendentalPredicatez Transcedental number predicate. Explanation =========== ``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. transcendentalTranscendentalz!Handler for Q.transcendental key.r Nr rrrr>r>~s  D 3Grr>N)sympy.assumptionsrsympy.multipledispatchrrrrrr"r&r*r.r2r6r:r>rrrrCs'-yB )   F")"J?I?DID2!y!H""JY6  Fir