\c@ s`ddlmZmZddlmZddlmZddlmZm Z m Z m Z m Z ddl mZddlmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlm Z m!Z!ddl"m#Z#m$Z$ddl%m&Z&m'Z'm(Z(ddl)m*Z*m+Z+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6ddl7m8Z8ddl9m:Z:m;Z;ddl<m=Z=m>Z>defdYZ?de?fdYZ@de?efdYZAde?eefdYZBd e?efd!YZCd"e?efd#YZDd$e e#e?fd%YZEd&e e#e?fd'YZFd(e?efd)YZGd*e,eH>> from sympy import Interval, FiniteSet >>> Interval(0, 1).union(Interval(2, 3)) Union(Interval(0, 1), Interval(2, 3)) >>> Interval(0, 1) + Interval(2, 3) Union(Interval(0, 1), Interval(2, 3)) >>> Interval(1, 2, True, True) + FiniteSet(2, 3) Union(Interval.Lopen(1, 2), {3}) Similarly it is possible to use the '-' operator for set differences: >>> Interval(0, 2) - Interval(0, 1) Interval.Lopen(1, 2) >>> Interval(1, 3) - FiniteSet(2) Union(Interval.Ropen(1, 2), Interval.Lopen(2, 3)) (tUnion(tselftother((s.lib/python2.7/site-packages/sympy/sets/sets.pytunionCscC s t||S(s Returns the intersection of 'self' and 'other'. >>> from sympy import Interval >>> Interval(1, 3).intersect(Interval(1, 2)) Interval(1, 2) >>> from sympy import imageset, Lambda, symbols, S >>> n, m = symbols('n m') >>> a = imageset(Lambda(n, 2*n), S.Integers) >>> a.intersect(imageset(Lambda(m, 2*m + 1), S.Integers)) EmptySet() (t Intersection(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt intersect^scC s |j|S(s/ Alias for :meth:`intersect()` (R8(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt intersectionpscC s|j|tjkS(sx Returns True if 'self' and 'other' are disjoint Examples ======== >>> from sympy import Interval >>> Interval(0, 2).is_disjoint(Interval(1, 2)) False >>> Interval(0, 2).is_disjoint(Interval(3, 4)) True References ========== .. [1] https://en.wikipedia.org/wiki/Disjoint_sets (R8RtEmptySet(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt is_disjointvscC s |j|S(s1 Alias for :meth:`is_disjoint()` (R;(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt isdisjointscC s t||S(s\ The complement of 'self' w.r.t the given universe. Examples ======== >>> from sympy import Interval, S >>> Interval(0, 1).complement(S.Reals) Union(Interval.open(-oo, 0), Interval.open(1, oo)) >>> Interval(0, 1).complement(S.UniversalSet) UniversalSet() \ Interval(0, 1) (t Complement(R4tuniverse((s.lib/python2.7/site-packages/sympy/sets/sets.pyt complementsc sttr^tdtjjD}d|D}tfd|DSttrttsttrtjt j Snttrtfdj DStt r t j dtj ddt Sttr"t jSttrdd lm}|fd }tt|t |drt t|ddt nt jSdS( Ncs s(|]\}}t|||VqdS(N(t FiniteSet(t.0tsto((s.lib/python2.7/site-packages/sympy/sets/sets.pys scs s|]}t|VqdS(N(t ProductSet(RAtset((s.lib/python2.7/site-packages/sympy/sets/sets.pys sc3 s!|]}|kr|VqdS(N((RAtp(R5(s.lib/python2.7/site-packages/sympy/sets/sets.pys sc3 s|]}|VqdS(N((RARC(R4(s.lib/python2.7/site-packages/sympy/sets/sets.pys siitevaluatei(R#c stj|S(N(Rtcontains(tx(R4(s.lib/python2.7/site-packages/sympy/sets/sets.pytt(t isinstanceRDtziptsetsR3tIntervalR@R7R?RtRealstargsR=tFalseR:tsympy.utilities.iterablesR#tNone(R4R5t switch_setst product_setsR#tsifted((R5R4s.lib/python2.7/site-packages/sympy/sets/sets.pyt _complements& *cC s t||S(sU Returns symmetric difference of `self` and `other`. Examples ======== >>> from sympy import Interval, S >>> Interval(1, 3).symmetric_difference(S.Reals) Union(Interval.open(-oo, 1), Interval.open(3, oo)) >>> Interval(1, 10).symmetric_difference(S.Reals) Union(Interval.open(-oo, 1), Interval.open(10, oo)) >>> from sympy import S, EmptySet >>> S.Reals.symmetric_difference(EmptySet()) Reals References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_difference (tSymmetricDifference(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pytsymmetric_differencescC stt||t||S(N(R3R=(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt_symmetric_differencescC s|jS(s The infimum of 'self' Examples ======== >>> from sympy import Interval, Union >>> Interval(0, 1).inf 0 >>> Union(Interval(0, 1), Interval(2, 3)).inf 0 (t_inf(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR)scC std|dS(Ns (%s)._inf(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR\scC s|jS(s The supremum of 'self' Examples ======== >>> from sympy import Interval, Union >>> Interval(0, 1).sup 1 >>> Union(Interval(0, 1), Interval(2, 3)).sup 3 (t_sup(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytsupscC std|dS(Ns (%s)._sup(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR] scC sOt|dt}t|j|}|dkrKt||dt}n|S(s@ Returns True if 'other' is contained in 'self' as an element. As a shortcut it is possible to use the 'in' operator: Examples ======== >>> from sympy import Interval >>> Interval(0, 1).contains(0.5) True >>> 0.5 in Interval(0, 1) True tstrictRGN(RtTruet _containsRTR!RR(R4R5tret((s.lib/python2.7/site-packages/sympy/sets/sets.pyRHs  cC std||fdS(Ns(%s)._contains(%s)(R,(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRa$scC s6t|tr"|j||kStd|dS(s# Returns True if 'self' is a subset of 'other'. Examples ======== >>> from sympy import Interval >>> Interval(0, 0.5).is_subset(Interval(0, 1)) True >>> Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) False sUnknown argument '%s'N(RLR(R8R.(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt is_subset'scC s |j|S(s/ Alias for :meth:`is_subset()` (Rc(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pytissubset:scC s<t|tr(||ko'|j|Std|dS(s( Returns True if 'self' is a proper subset of 'other'. Examples ======== >>> from sympy import Interval >>> Interval(0, 0.5).is_proper_subset(Interval(0, 1)) True >>> Interval(0, 1).is_proper_subset(Interval(0, 1)) False sUnknown argument '%s'N(RLR(RcR.(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_proper_subset@scC s0t|tr|j|Std|dS(s) Returns True if 'self' is a superset of 'other'. Examples ======== >>> from sympy import Interval >>> Interval(0, 0.5).is_superset(Interval(0, 1)) False >>> Interval(0, 1).is_superset(Interval(0, 1, left_open=True)) True sUnknown argument '%s'N(RLR(RcR.(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt is_supersetSs cC s |j|S(s1 Alias for :meth:`is_superset()` (Rf(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt issupersetfscC s<t|tr(||ko'|j|Std|dS(s. Returns True if 'self' is a proper superset of 'other'. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).is_proper_superset(Interval(0, 0.5)) True >>> Interval(0, 1).is_proper_superset(Interval(0, 1)) False sUnknown argument '%s'N(RLR(RfR.(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_proper_supersetlscC std|jdS(NsPower set not defined for: %s(R,tfunc(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyt_eval_powersetscC s |jS(s Find the Power set of 'self'. Examples ======== >>> from sympy import FiniteSet, EmptySet >>> A = EmptySet() >>> A.powerset() {EmptySet()} >>> A = FiniteSet(1, 2) >>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2) >>> A.powerset() == FiniteSet(a, b, c, EmptySet()) True References ========== .. [1] https://en.wikipedia.org/wiki/Power_set (Rj(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytpowersetscC s|jS(s The (Lebesgue) measure of 'self' Examples ======== >>> from sympy import Interval, Union >>> Interval(0, 1).measure 1 >>> Union(Interval(0, 1), Interval(2, 3)).measure 2 (t_measure(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytmeasurescC s|jS(s= The boundary or frontier of a set A point x is on the boundary of a set S if 1. x is in the closure of S. I.e. Every neighborhood of x contains a point in S. 2. x is not in the interior of S. I.e. There does not exist an open set centered on x contained entirely within S. There are the points on the outer rim of S. If S is open then these points need not actually be contained within S. For example, the boundary of an interval is its start and end points. This is true regardless of whether or not the interval is open. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).boundary {0, 1} >>> Interval(0, 1, True, False).boundary {0, 1} (t _boundary(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytboundaryscC st||jstSdS(s  Property method to check whether a set is open. A set is open if and only if it has an empty intersection with its boundary. Examples ======== >>> from sympy import S >>> S.Reals.is_open True N(R7RoR`RT(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_opens cC s|jj|S(s A property method to check whether a set is closed. A set is closed if it's complement is an open set. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).is_closed True (RoRc(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyt is_closeds cC s ||jS(sN Property method which returns the closure of a set. The closure is defined as the union of the set itself and its boundary. Examples ======== >>> from sympy import S, Interval >>> S.Reals.closure Reals >>> Interval(0, 1).closure Interval(0, 1) (Ro(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytclosurescC s ||jS(sz Property method which returns the interior of a set. The interior of a set S consists all points of S that do not belong to the boundary of S. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).interior Interval.open(0, 1) >>> Interval(0, 1).boundary.interior EmptySet() (Ro(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytinteriorscC s tdS(N(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRn scC std|dS(Ns (%s)._measure(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRlscC s |j|S(N(R6(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__add__scC s |j|S(N(R6(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__or__scC s |j|S(N(R8(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__and__scC s t||S(N(RD(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__mul__scC s t||S(N(RY(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__xor__scC s@t|j r/|dkr/td|nt|g|S(Nis'%s: Exponent must be a positive Integer(Rt is_IntegerR.RD(R4texp((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__pow__"scC s t||S(N(R=(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__sub__'scC sPt|j|}|tjkp0|tjksFtd|nt|S(Ns'contains did not evaluate to a bool: %r(RRHRRR t TypeErrortbool(R4R5tsymb((s.lib/python2.7/site-packages/sympy/sets/sets.pyt __contains__*sN(;t__name__t __module__t__doc__RRt is_numbert is_iterablet is_intervalt is_FiniteSett is_Intervalt is_ProductSettis_UnionRTtis_Intersectiont is_EmptySettis_UniversalSett is_Complementtis_ComplexRegiont staticmethodR2R6R8R9R;R<R?RXRZR[tpropertyR)R\R^R]RHRaRcRdReRfRgRhRjRkRmRoRpRqRrRsRnRlRtRuRvRwRxR{R|R(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR(sj        (                   RDcB seZdZeZdZdZdZedZ edZ edZ dZ edZ d Zd ZeZRS( sS Represents a Cartesian Product of Sets. Returns a Cartesian product given several sets as either an iterable or individual arguments. Can use '*' operator on any sets for convenient shorthand. Examples ======== >>> from sympy import Interval, FiniteSet, ProductSet >>> I = Interval(0, 5); S = FiniteSet(1, 2, 3) >>> ProductSet(I, S) Interval(0, 5) x {1, 2, 3} >>> (2, 2) in ProductSet(I, S) True >>> Interval(0, 1) * Interval(0, 1) # The unit square Interval(0, 1) x Interval(0, 1) >>> coin = FiniteSet('H', 'T') >>> set(coin**2) {(H, H), (H, T), (T, H), (T, T)} Notes ===== - Passes most operations down to the argument sets - Flattens Products of ProductSets References ========== .. [1] https://en.wikipedia.org/wiki/Cartesian_product c svfdt|}t|ksBt|dkrItSt|dkrc|dStj|||S(Nc smt|tr;|jr1tt|jgS|gSn"t|r]tt|gStddS(Ns'Input must be Sets or iterables of Sets(RLR(RtsumtmapRQRR}(targ(tflatten(s.lib/python2.7/site-packages/sympy/sets/sets.pyR[s   ii(tlistR:tlenRt__new__(tclsRNt assumptions((Rs.lib/python2.7/site-packages/sympy/sets/sets.pyRZs !cC sR|js dSt|jt|jkr/tStdt|j|jDS(Ncs s$|]\}}t||VqdS(N(R(RARIty((s.lib/python2.7/site-packages/sympy/sets/sets.pys us(RRRQR RRM(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt_eval_Eqns  cC spy#t|t|jkr"tSWntk r7tSXtgt|j|D]\}}|j|^qNS(s5 'in' operator for ProductSets Examples ======== >>> from sympy import Interval >>> (2, 3) in Interval(0, 5) * Interval(0, 5) True >>> (10, 10) in Interval(0, 5) * Interval(0, 5) False Passes operation on to constituent sets (RRQR R}RRMRNRH(R4telementREtitem((s.lib/python2.7/site-packages/sympy/sets/sets.pyRaws cC s|jS(N(RQ(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRNsc s#tfdtjDS(Nc3 s:|]0\}tfdtjDVqdS(c3 s7|]-\}}|kr(||jn|jVqdS(N(Ro(RAtjtb(ti(s.lib/python2.7/site-packages/sympy/sets/sets.pys sN(RDt enumerateRN(RAta(R4(Rs.lib/python2.7/site-packages/sympy/sets/sets.pys s(R3RRN(R4((R4s.lib/python2.7/site-packages/sympy/sets/sets.pyRnscC std|jDS(s A property method which tests whether a set is iterable or not. Returns True if set is iterable, otherwise returns False. Examples ======== >>> from sympy import FiniteSet, Interval, ProductSet >>> I = Interval(0, 1) >>> A = FiniteSet(1, 2, 3, 4, 5) >>> I.is_iterable False >>> A.is_iterable True cs s|]}|jVqdS(N(R(RARE((s.lib/python2.7/site-packages/sympy/sets/sets.pys s(tallRN(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s&|jrt|jStddS(s A method which implements is_iterable property method. If self.is_iterable returns True (both constituent sets are iterable), then return the Cartesian Product. Otherwise, raise TypeError. s%Not all constituent sets are iterableN(RRRNR}(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__iter__s  cC s+d}x|jD]}||j9}qW|S(Ni(RNRm(R4RmRE((s.lib/python2.7/site-packages/sympy/sets/sets.pyRlscC s&tg|jD]}t|^q S(N(RRQR(R4RB((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__len__scC s&tg|jD]}t|^q S(N(RRQR~(R4RB((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__bool__s(RRRR`RRRRaRRNRnRRRlRRt __nonzero__(((s.lib/python2.7/site-packages/sympy/sets/sets.pyRD1s&     ROcB seZdZeZeedZedZeZ Z e dZ e dZ e dZedZeZZedZedZd Zed Zd Zed Zd dZdZdZedZedZdZdZRS(s Represents a real interval as a Set. Usage: Returns an interval with end points "start" and "end". For left_open=True (default left_open is False) the interval will be open on the left. Similarly, for right_open=True the interval will be open on the right. Examples ======== >>> from sympy import Symbol, Interval >>> Interval(0, 1) Interval(0, 1) >>> Interval.Ropen(0, 1) Interval.Ropen(0, 1) >>> Interval.Ropen(0, 1) Interval.Ropen(0, 1) >>> Interval.Lopen(0, 1) Interval.Lopen(0, 1) >>> Interval.open(0, 1) Interval.open(0, 1) >>> a = Symbol('a', real=True) >>> Interval(0, a) Interval(0, a) Notes ===== - Only real end points are supported - Interval(a, b) with a > b will return the empty set - Use the evalf() method to turn an Interval into an mpmath 'mpi' interval instance References ========== .. [1] https://en.wikipedia.org/wiki/Interval_%28mathematics%29 c st|}t|}t|}t|}td||gDsetd||fntjtjgtfd||fDstdn||ktkrtjS||j rtjS||kr|s|rtjS||kr<|p | r<|tjks+|tjkr2tjSt |S|tjkrTt }n|tjkrlt }nt j |||||S(Ncs s0|]&}t|ttttfVqdS(N(RLttypeRR (RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys ssGleft_open and right_open can have only true/false values, got %s and %sc3 s*|] }|jtk p!|kVqdS(N(tis_realRR(RAR(tinftys(s.lib/python2.7/site-packages/sympy/sets/sets.pys ss$Non-real intervals are not supported(RRR,RR/tNegativeInfinityR.R`R:t is_negativeR@RRR(Rtstarttendt left_opent right_open((Rs.lib/python2.7/site-packages/sympy/sets/sets.pyRs6     "    cC s |jdS(s The left end point of 'self'. This property takes the same value as the 'inf' property. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).start 0 i(t_args(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s|||ttS(s.Return an interval including neither boundary.(R`(RRR((s.lib/python2.7/site-packages/sympy/sets/sets.pytopen/scC s|||ttS(s3Return an interval not including the left boundary.(R`RR(RRR((s.lib/python2.7/site-packages/sympy/sets/sets.pytLopen4scC s|||ttS(s4Return an interval not including the right boundary.(RRR`(RRR((s.lib/python2.7/site-packages/sympy/sets/sets.pytRopen9scC s |jdS(s The right end point of 'self'. This property takes the same value as the 'sup' property. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).end 1 i(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR>scC s |jdS(s True if 'self' is left-open. Examples ======== >>> from sympy import Interval >>> Interval(0, 1, left_open=True).left_open True >>> Interval(0, 1, left_open=False).left_open False i(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRQscC s |jdS(s True if 'self' is right-open. Examples ======== >>> from sympy import Interval >>> Interval(0, 1, right_open=True).right_open True >>> Interval(0, 1, right_open=False).right_open False i(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRbscC s|tjkrZttj|jt|j }t|jtj|j t}t ||St |t rg|j D]}|jrs|^qs}|gkrdSntj||S(N(RRPRORRR`RRR/RR3RLR@RQRRTR(RX(R4R5RRtmtnums((s.lib/python2.7/site-packages/sympy/sets/sets.pyRXss % cC sDg|j|jfD]!}t|tjkr|^q}t|S(N(RRtabsRR/R@(R4RFt finite_points((s.lib/python2.7/site-packages/sympy/sets/sets.pyRns!cC s t|t s[|tjks[|tjks[|tjks[|tjks[|jtkr_t S|j tjkr|j tjkr|jdk r|jSn|j r||j k}n||j k}|jrt|||j k}nt|||j k}t|S(N(RLR RR/RtNaNtComplexInfinityRRRR RRRTRRRR(R4R5R0((s.lib/python2.7/site-packages/sympy/sets/sets.pyRas $   cC s|j|jS(N(RR(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRlsi5cC s1tt|jj|t|jj|S(N(R&R'Rt _eval_evalfR(R4tprec((s.lib/python2.7/site-packages/sympy/sets/sets.pytto_mpiscC s7t|jj||jj|d|jd|jS(NRR(ROtleftRtrightRR(R4R((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s@|jj}||jjM}||jjM}||jjM}|S(N(RR*R(R4R5R*((s.lib/python2.7/site-packages/sympy/sets/sets.pyt_is_comparables  cC s%|jtjkp$|jtdkS(s;Return ``True`` if the left endpoint is negative infinity. s-inf(RRRR(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_left_unboundedscC s%|jtjkp$|jtdkS(s<Return ``True`` if the right endpoint is positive infinity. s+inf(RRR/R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_right_unboundedscC smt|}|jr'||jk}n||jk}|jrQ|j|k}n|j|k}t||S(sARewrite an interval in terms of inequalities and logic operators.(RRRRRR(R4RIRR((s.lib/python2.7/site-packages/sympy/sets/sets.pyt as_relationals   cC st|ts9t|tr"tSt|tr5dStStt|j|jt|j |j |j |j k|j |j kS(N( RLROR@R R(RTRRRRRR(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs(RRRR`RRRRRRR\Rt classmethodRRRRR]RRRRXRnRaRlRRRRRRR(((s.lib/python2.7/site-packages/sympy/sets/sets.pyROs.)'        R3cB seZdZeZedZedZdZee dZ dZ edZ edZ dZed Zed Zd Zed Zd ZdZRS(s  Represents a union of sets as a :class:`Set`. Examples ======== >>> from sympy import Union, Interval >>> Union(Interval(1, 2), Interval(3, 4)) Union(Interval(1, 2), Interval(3, 4)) The Union constructor will always try to merge overlapping intervals, if possible. For example: >>> Union(Interval(1, 2), Interval(2, 3)) Interval(1, 3) See Also ======== Intersection References ========== .. [1] https://en.wikipedia.org/wiki/Union_%28set_theory%29 cC stjS(N(RR:(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytidentityscC stjS(N(Rt UniversalSet(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pytzeroscO s|jdtd}t|}|rGt|j|}t|Stt|tj}t j ||}t ||_ |S(NRGi( tgetR RRt_new_args_filtertsimplify_unionRR(R2RRt frozensett_argset(RRQtkwargsRGtobj((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs  cC s|jS(N(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRQsc stfd|jDS(Nc3 s|]}|jVqdS(N(R?(RARB(R>(s.lib/python2.7/site-packages/sympy/sets/sets.pys s(R7RQ(R4R>((R>s.lib/python2.7/site-packages/sympy/sets/sets.pyRXscC s3ddlm}|g|jD]}|j^qS(Ni(tMin(t(sympy.functions.elementary.miscellaneousRRQR)(R4RRE((s.lib/python2.7/site-packages/sympy/sets/sets.pyR\scC s3ddlm}|g|jD]}|j^qS(Ni(tMax(RRRQR^(R4RRE((s.lib/python2.7/site-packages/sympy/sets/sets.pyR]scC s)tg|jD]}|j|^q S(N(RRQRH(R4R5RE((s.lib/python2.7/site-packages/sympy/sets/sets.pyRa&sc  sgjD]}t||f^q }d}d}x|r ||td|D7}fd|D}g|D]'\}}|jdkrx||f^qx}g}g}xB|D]:} | d|krqq|j| d|j| qW|}|d9}q7W|S(Niics s|]\}}|jVqdS(N(Rm(RAtsostinter((s.lib/python2.7/site-packages/sympy/sets/sets.pys ;sc3 sP|]F\}}jD]0}||kr|t||j|fVqqdS(N(RQR@R8(RARR9tnewset(R4(s.lib/python2.7/site-packages/sympy/sets/sets.pys ?si(RQR@RRmtappend( R4RBRNRmtparityRRtsos_listt sets_listRE((R4s.lib/python2.7/site-packages/sympy/sets/sets.pyRl)s$ (   4 c s1fd}tt|ttjS(Nc sPj|j}x9tjD](\}}||kr ||j}q q W|S(s8 The boundary of set i minus interior of all other sets (RQRoRRs(RRRR(R4(s.lib/python2.7/site-packages/sympy/sets/sets.pytboundary_of_setWs  (R3RRRRQ(R4R((R4s.lib/python2.7/site-packages/sympy/sets/sets.pyRnUscC st|jdkr|j\}}|j|jkr|jtjkr|jtjkrtt||j||jk||jkSnt g|jD]}|j |^qS(s<Rewrite a Union in terms of equalities and logic operators. i( RRQR^R)RRR/RRRR(R4tsymbolRRRE((s.lib/python2.7/site-packages/sympy/sets/sets.pyR`s $1cC std|jDS(Ncs s|]}|jVqdS(N(R(RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys ks(RRQ(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRisc smy!tfd|jDSWnEtttfk rhddl}tdd|jdfnXdS(Nc3 s|]}|jVqdS(N(R(RARE(R(s.lib/python2.7/site-packages/sympy/sets/sets.pys osisNot all sets are evalf-ablei(R3RQR}R.R,tsysRTtexc_info(R4RR((Rs.lib/python2.7/site-packages/sympy/sets/sets.pyRms!  c s[ddlfd}td|jDrK|d|jDStddS(Nic7 st|}tr.jd|D}njd|D}x`|ryx|D]}|VqZWWqJtk r|d8}jj||}qJXqJWdS(s,roundrobin('ABC', 'D', 'EF') --> A D E B F Ccs s|]}t|jVqdS(N(titert__next__(RAtit((s.lib/python2.7/site-packages/sympy/sets/sets.pys scs s|]}t|jVqdS(N(Rtnext(RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys siN(RRtcyclet StopIterationtislice(t iterablestpendingtnextsR(t itertools(s.lib/python2.7/site-packages/sympy/sets/sets.pyt roundrobin{s     cs s|]}|jVqdS(N(R(RARE((s.lib/python2.7/site-packages/sympy/sets/sets.pys scs s|]}t|VqdS(N(R(RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys ss%Not all constituent sets are iterable(RRRQR}(R4R((Rs.lib/python2.7/site-packages/sympy/sets/sets.pyRvs  (RRRR`RRRRRR RQRXR\R]RaRlRnRRRR(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR3s"   ,  R7cB seZdZeZedZedZdZee dZ edZ edZ edZ dZd Zed Zd ZRS( s Represents an intersection of sets as a :class:`Set`. Examples ======== >>> from sympy import Intersection, Interval >>> Intersection(Interval(1, 3), Interval(2, 4)) Interval(2, 3) We often use the .intersect method >>> Interval(1,3).intersect(Interval(2,4)) Interval(2, 3) See Also ======== Union References ========== .. [1] https://en.wikipedia.org/wiki/Intersection_%28set_theory%29 cC stjS(N(RR(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC stjS(N(RR:(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscO s|jdtd}t|}|rGt|j|}t|Stt|tj}t j ||}t ||_ |S(NRGi( RR RRRtsimplify_intersectionRR(R2RRRR(RRQRRGR((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs  cC s|jS(N(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRQscC std|jDS(Ncs s|]}|jVqdS(N(R(RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys s(tanyRQ(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s tdS(N(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR\scC s tdS(N(R,(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyR]scC s)tg|jD]}|j|^q S(N(RRQRH(R4R5RE((s.lib/python2.7/site-packages/sympy/sets/sets.pyRascc st}x|jD]}|jrt}t|jt|f}tdt|}xT|D]I}t|j|}|tj kr|VqZ|tj krqZ|VqZWqqW|rt dndS(NRGs)None of the constituent sets are iterable( R`RQRRRRER7RRHRRR R.(R4tno_iterRBt other_setsR5RItc((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs  c sddlm}mddlm}t|ddt\}}|sNdS|jdt|d}|d}g}g}xc|D][|fd ||D}|r|j q|dkr|j qqW|rt d t |nt j}|r{g|D]jtr^q} |t d t | } x|r|j} td || d | DDrx| D]"|kr|jqqWqLg| D]*t| jt jkr^q} | | krL|j | t d t | qLqLWt|} | s7t jS| t jkrY|t |7}q{|tt || d t 7}n|S( Ni(t fuzzy_andR(t zip_longestcS s|jS(N(R(RI((s.lib/python2.7/site-packages/sympy/sets/sets.pyRJRKtbinarytkeyiic3 s$|]}|jVqdS(N(RH(RARC(RRI(s.lib/python2.7/site-packages/sympy/sets/sets.pys sRGcs s!|]\}}||kVqdS(N((RARR((s.lib/python2.7/site-packages/sympy/sets/sets.pys scs s$|]}|jtr|VqdS(N(thasR(RARI((s.lib/python2.7/site-packages/sympy/sets/sets.pys s(tsympy.core.logicRRtsympy.core.compatibilityRR#R`tsortRRRTR@RRRR:RRtpopRtremoveRRHRR7R(RQRRtfs_argsR5RBtrestunkRtsymbolic_s_listtnon_symbolic_stvt containedR((RRIs.lib/python2.7/site-packages/sympy/sets/sets.pyt_handle_finite_setss`    !(        *    cC s)tg|jD]}|j|^q S(sBRewrite an Intersection in terms of equalities and logic operators(RRQR(R4RRE((s.lib/python2.7/site-packages/sympy/sets/sets.pyR/s(RRRR`RRRRRR RQRR\R]RaRRRR(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR7s   CR=cB s8eZdZeZedZedZdZRS(sRepresents the set difference or relative complement of a set with another set. `A - B = \{x \in A| x \\notin B\}` Examples ======== >>> from sympy import Complement, FiniteSet >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) {0, 2} See Also ========= Intersection, Union References ========== .. [1] http://mathworld.wolfram.com/ComplementSet.html cC s)|rtj||Stj|||S(N(R=treduceRR(RRRRG((s.lib/python2.7/site-packages/sympy/sets/sets.pyROsc s|tjksj|r%tSt|trQtfd|jDS|j}|dk rp|St |dt SdS(s2 Simplify a :class:`Complement`. c3 s|]}|jVqdS(N(R?(RARB(tA(s.lib/python2.7/site-packages/sympy/sets/sets.pys _sRGN( RRRcR:RLR3R7RQRXRTR=RR(RtBtresult((Rs.lib/python2.7/site-packages/sympy/sets/sets.pyRUs cC s?|jd}|jd}t|j|t|j|S(Nii(RQRRHR(R4R5RR((s.lib/python2.7/site-packages/sympy/sets/sets.pyRags  ( RRRR`RRRRRa(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR=4s  R:cB sweZdZeZeZedZdZdZ dZ dZ dZ edZ dZd ZRS( s Represents the empty set. The empty set is available as a singleton as S.EmptySet. Examples ======== >>> from sympy import S, Interval >>> S.EmptySet EmptySet() >>> Interval(1, 2).intersect(S.EmptySet) EmptySet() See Also ======== UniversalSet References ========== .. [1] https://en.wikipedia.org/wiki/Empty_set cC sdS(Ni((R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRlscC stS(N(R (R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRascC stS(N(R (R4R((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC sdS(Ni((R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s tgS(N(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s t|S(N(R@(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRjscC s|S(N((R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRnscC s|S(N((R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRXscC s|S(N((R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyR[s(RRRR`RRRRlRaRRRRjRnRXR[(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR:ms      RcB sVeZdZeZdZdZedZdZ dZ edZ RS(s Represents the set of all things. The universal set is available as a singleton as S.UniversalSet Examples ======== >>> from sympy import S, Interval >>> S.UniversalSet UniversalSet() >>> Interval(1, 2).intersect(S.UniversalSet) Interval(1, 2) See Also ======== EmptySet References ========== .. [1] https://en.wikipedia.org/wiki/Universal_set cC stjS(N(RR:(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRXscC s|S(N((R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyR[scC stjS(N(RR/(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRlscC stS(N(R(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRascC stS(N(R(R4R((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC stS(N(R:(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRns( RRRR`RRXR[RRlRaRRn(((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs    R@cB seZdZeZeZdZdZdZdZ dZ e dZ e dZ e dZe d Zd Zd Zd Zd ZdZe dZdZdZdZdZdZRS(s Represents a finite set of discrete numbers Examples ======== >>> from sympy import FiniteSet >>> FiniteSet(1, 2, 3, 4) {1, 2, 3, 4} >>> 3 in FiniteSet(1, 2, 3, 4) True >>> members = [1, 2, 3, 4] >>> f = FiniteSet(*members) >>> f {1, 2, 3, 4} >>> f - FiniteSet(2) {1, 3, 4} >>> f + FiniteSet(2, 5) {1, 2, 3, 4, 5} References ========== .. [1] https://en.wikipedia.org/wiki/Finite_set cO s|jdtd}|rMttt|}t|dkrbtSnttt|}tttt |t j }t j ||}t||_|S(NRGi(RR RRRRR:RRttupleR(R2RRt _elements(RRQRRGR((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs $cC sxt|ts9t|tr"tSt|tr5dStSt|t|krUtStdt|j |j DS(Ncs s$|]\}}t||VqdS(N(R(RARIR((s.lib/python2.7/site-packages/sympy/sets/sets.pys s( RLR@ROR R(RTRRRMRQ(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s t|jS(N(RRQ(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRsc C sBt|trOtd|jD}|tjkr<|gkr<g|jD]}|jrM|^qM}g}|ttj|dttg7}xAt |d |dD](\}}|j t||ttqW|j t|dtj tt|gkr)t t dt|t|dtSt dt|Sq2|gkr2dSnt|tr2g}xQ|D]I} t|j| } | tjk rk| tjk rk|j | qkqkWt|}||krdSg} xB|D]:} t|j| } | tjk r| j | qqWt t| |Stj||S(Ncs s|]}|jr|VqdS(N(R(RAR((s.lib/python2.7/site-packages/sympy/sets/sets.pys siiiRG(RLROtsortedRQRRPt is_SymbolRR`RMRR/R=R3RRR@RTRRHRR R(RX( R4R5RRtsymst intervalsRRRRRtnot_true((s.lib/python2.7/site-packages/sympy/sets/sets.pyRXs>%#$ #      cC sXt}xK|jD]@}t||dt}|tkr;|S|tk rd}qqW|S(sR Tests whether an element, other, is in the set. Relies on Python's set class. This tests for object equality All inputs are sympified Examples ======== >>> from sympy import FiniteSet >>> 1 in FiniteSet(1, 2) True >>> 5 in FiniteSet(1, 2) False RGN(R RRR`RRT(R4R5trtett((s.lib/python2.7/site-packages/sympy/sets/sets.pyRa<s   cC s|S(N((R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRnXscC sddlm}||S(Ni(R(RR(R4R((s.lib/python2.7/site-packages/sympy/sets/sets.pyR\\scC sddlm}||S(Ni(R(RR(R4R((s.lib/python2.7/site-packages/sympy/sets/sets.pyR]ascC sdS(Ni((R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRmfscC s t|jS(N(RRQ(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyRjscC s6ddlm}tg|D]}|||^qS(s@Rewrite a FiniteSet in terms of equalities and logic operators. i(R(tsympy.core.relationalRR(R4RRtelem((s.lib/python2.7/site-packages/sympy/sets/sets.pyRmscC st|t|S(N(thash(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pytcomparerscC s&tg|D]}|j|^q S(N(R@R(R4RR((s.lib/python2.7/site-packages/sympy/sets/sets.pyRuscC s |jfS(N(R(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyt_hashable_contentxscC stt|jtjS(N(RRRQR(R2(R4((s.lib/python2.7/site-packages/sympy/sets/sets.pyt _sorted_args{scC s2|jgt|jD]}|j|^qS(N(RiR"RQ(R4RB((s.lib/python2.7/site-packages/sympy/sets/sets.pyRjscC s5t|ts(tdt|n|j|S(Ns!Invalid comparison of set with %s(RLR(R}R$Rc(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__ge__scC s5t|ts(tdt|n|j|S(Ns!Invalid comparison of set with %s(RLR(R}R$Rh(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__gt__scC s5t|ts(tdt|n|j|S(Ns!Invalid comparison of set with %s(RLR(R}R$Rc(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__le__scC s5t|ts(tdt|n|j|S(Ns!Invalid comparison of set with %s(RLR(R}R$Re(R4R5((s.lib/python2.7/site-packages/sympy/sets/sets.pyt__lt__s(RRRR`RRRRRRXRaRRnR\R]RmRRRRRRRjRRRR(((s.lib/python2.7/site-packages/sympy/sets/sets.pyR@s.   '          cC s t|S(N(R@(RI((s.lib/python2.7/site-packages/sympy/sets/sets.pyRJRKcC s t|S(N(R@(RI((s.lib/python2.7/site-packages/sympy/sets/sets.pyRJRKRYcB s/eZdZeZedZedZRS(sRepresents the set of elements which are in either of the sets and not in their intersection. Examples ======== >>> from sympy import SymmetricDifference, FiniteSet >>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5)) {1, 2, 4, 5} See Also ======== Complement, Union References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_difference cC s)|rtj||Stj|||S(N(RYRRR(RRRRG((s.lib/python2.7/site-packages/sympy/sets/sets.pyRscC s6|j|}|dk r|St||dtSdS(NRG(R[RTRYRR(RRR((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs (RRRR`tis_SymmetricDifferenceRRR(((s.lib/python2.7/site-packages/sympy/sets/sets.pyRYs c G sddlm}ddlm}ddlm}t|dkr[tdt|nt|dt t frt|dkr||d|d}|d}n|d}|d}t||rnt|t st |d krt|dkr0t t d |t}||}n?gtt|D]}t d |d^qC}||}|||}nttd t |td |Drg|D]} t | ^q} td| nt|dkr|d} ||| } | dkr.||| } nt| |rO| j\}} n|jd|jkri| St| |rt| jjdkrt|jdkrt|| jjd|jj|jd| jj| jSn| dk r| Sn|||S(s{ Return an image of the set under transformation ``f``. If this function can't compute the image, it returns an unevaluated ImageSet object. .. math:: { f(x) | x \in self } Examples ======== >>> from sympy import S, Interval, Symbol, imageset, sin, Lambda >>> from sympy.abc import x, y >>> imageset(x, 2*x, Interval(0, 2)) Interval(0, 4) >>> imageset(lambda x: 2*x, Interval(0, 2)) Interval(0, 4) >>> imageset(Lambda(x, sin(x)), Interval(-2, 1)) ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) >>> imageset(sin, Interval(-2, 1)) ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) >>> imageset(lambda y: x + y, Interval(-2, 1)) ImageSet(Lambda(_x, _x + x), Interval(-2, 1)) Expressions applied to the set of Integers are simplified to show as few negatives as possible and linear expressions are converted to a canonical form. If this is not desirable then the unevaluated ImageSet should be used. >>> imageset(x, -2*x + 5, S.Integers) ImageSet(Lambda(x, 2*x + 1), Integers) See Also ======== sympy.sets.fancysets.ImageSet i(tLambda(tImageSet(t set_functionis)imageset expects at least 2 args, got: %siisRIsx%is> expecting lambda, Lambda, or FunctionClass, not '%s'.cs s|]}t|t VqdS(N(RLR((RARB((s.lib/python2.7/site-packages/sympy/sets/sets.pys ss.arguments after mapping should be sets, not %sN(t sympy.coreRtsympy.sets.fancysetsRtsympy.sets.setexprRRR.RLRRR R$RRRR}R%RRTRQt variablesR0tlamdatimagesettsubstbase_set( RQRRRtftset_listtvarR0RRBtnameRER ((s.lib/python2.7/site-packages/sympy/sets/sets.pyR"sX,+   3   -"  cC syddlm}m}|||fkr,tStd}||j|}|dktksq|dktkrutSdS(sk Checks whether function ``func`` is invertible when the domain is restricted to set ``setv``. i(RztlogtuiN(tsympyRzR)R`RtdiffRT(RitsetvRzR)R*tfdiff((s.lib/python2.7/site-packages/sympy/sets/sets.pytis_function_invertible_in_set+s $c C sddlm}|stjSx,|D]$}t|ts$tdq$q$Wg|D]}|jrS|^qS}t|dkrd|D}t |}|gg|D]}|js|^q}nt |}t }x|rx|D]}t }xz|t |fD]e} ||| } | dk r t| t sLt | f} n|t || fj| }Pq q W|r|}PqqWqWt|dkr|jStdt |SdS(s8 Simplify a :class:`Union` using known rules We first start with global rules like 'Merge all FiniteSets' Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent. This process depends on ``union_sets(a, b)`` functions. i(t union_setss Input args to Union must be Setsics s"|]}|D] }|Vq qdS(N((RARERI((s.lib/python2.7/site-packages/sympy/sets/sets.pys UsRGN(tsympy.sets.handlers.unionR0RR:RLR(R}RRR@RER`RRRTR6RR3( RQR0RRIt finite_setsRt finite_settnew_argsRBRtnew_set((s.lib/python2.7/site-packages/sympy/sets/sets.pyR>s:  " ,      c  sI|s tjSx,|D]$}t|tstdqqWtd|DrYtjStj|}|d k rx|Sx|D]}|j rt |t |f}t |dkrt|t fd|jDSt g|jD] }|^qSqqWxR|D]J}|jr|j|||jdg}tt||jdSqWddlm}t |}t}x|rx|D]}t}xb|t |fD]M}|||}|d k r|t ||fjt |f}PqqW|r|}PqqWqWt |dkr5|jStdt|Sd S( s' Simplify an intersection using known rules We first start with global rules like 'if any empty sets return empty set' and 'distribute any unions' Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent s Input args to Union must be Setscs s|]}|jVqdS(N(R(RARB((s.lib/python2.7/site-packages/sympy/sets/sets.pys sic3 s|]}t|VqdS(N(R7(RAR(R5(s.lib/python2.7/site-packages/sympy/sets/sets.pys sii(tintersection_setsRGN(RRRLR(R}RR:R7RRTRRERR3RQRRR=t sympy.sets.handlers.intersectionR6R`RRR6R( RQRtrvRBRR6R4RR5((R5s.lib/python2.7/site-packages/sympy/sets/sets.pyRssN      '       (  c C st||gddt\}}t|dkrqtg|dD]&}|dD]}|||^qRqDSt|dkrg|dD]}t||d||^q}t|SdSdS(NcS s t|tS(N(RLR@(RI((s.lib/python2.7/site-packages/sympy/sets/sets.pyRJRKRiii(R#R`RR@t_apply_operationR3RT( topRIRt commutativeRR5RRRN((s.lib/python2.7/site-packages/sympy/sets/sets.pyRs$;0 c C sjddlm}ddlm}m}td}t||||}|dkre|||}n|dkr|r|||}n|dkrf|d\} } t|t rt|t  r|||||||j }qft|t  r9t|t r9|||||||j }qf||| | f|| | ||}n|S(Ni(R(tsymbolsRtdsx y( t sympy.setsRR+R<RRRRTRLR(tdoit( R:RIRR;RR<RR=toutt_xt_y((s.lib/python2.7/site-packages/sympy/sets/sets.pyR9s    **-cC s&ddlm}t|||dtS(Ni(t_set_addR;(tsympy.sets.handlers.addRCR9R`(RIRRC((s.lib/python2.7/site-packages/sympy/sets/sets.pytset_addscC s&ddlm}t|||dtS(Ni(t_set_subR;(RDRFR9RR(RIRRF((s.lib/python2.7/site-packages/sympy/sets/sets.pytset_subscC s&ddlm}t|||dtS(Ni(t_set_mulR;(tsympy.sets.handlers.mulRHR9R`(RIRRH((s.lib/python2.7/site-packages/sympy/sets/sets.pytset_mulscC s&ddlm}t|||dtS(Ni(t_set_divR;(RIRKR9RR(RIRRK((s.lib/python2.7/site-packages/sympy/sets/sets.pytset_divscC s&ddlm}t|||dtS(Ni(t_set_powR;(tsympy.sets.handlers.powerRMR9RR(RIRRM((s.lib/python2.7/site-packages/sympy/sets/sets.pytset_powscC sddlm}|||S(Ni(t _set_function(tsympy.sets.handlers.functionsRP(R%RIRP((s.lib/python2.7/site-packages/sympy/sets/sets.pyRsN(Wt __future__RRRRtsympy.core.basicRRRRRRRtsympy.core.cacheR tsympy.core.evalfR tsympy.core.evaluateR tsympy.core.exprR tsympy.core.functionR RRtsympy.core.mulRtsympy.core.numbersRtsympy.core.operationsRRRRtsympy.core.singletonRRtsympy.core.symbolRRRtsympy.core.sympifyRRRtsympy.logic.boolalgRRRRR tsympy.sets.containsR!tsympy.utilitiesR"RSR#tsympy.utilities.miscR$R%tmpmathR&R'R(RDROR3R7R=R:RR@RERRYR"R/RRRR9RERGRJRLROR(((s.lib/python2.7/site-packages/sympy/sets/sets.pytsb((9:1  ' i  5 L