  
  [1X20 [33X[0;0YBooleans[133X[101X
  
  [33X[0;0YThe  two  main [13Xboolean[113X values are [9Xtrue[109X and [9Xfalse[109X. They stand for the [13Xlogical[113X
  values  of  the  same  name.  They  appear  as  values  of the conditions in
  [9Xif[109X-statements  and [9Xwhile[109X-loops. Booleans are also important as return values
  of  [13Xfilters[113X  (see [14X13.2[114X)  such as [2XIsFinite[102X ([14X30.4-2[114X) and [2XIsBool[102X ([14X20.1-1[114X). Note
  that  it  is  a  convention that the name of a function that returns [9Xtrue[109X or
  [9Xfalse[109X according to the outcome, starts with [10XIs[110X.[133X
  
  [33X[0;0YFor  technical  reasons,  also  the  value  [9Xfail[109X (see [14X20.2[114X) is regarded as a
  boolean.[133X
  
  
  [1X20.1 [33X[0;0YIsBool (Filter)[133X[101X
  
  [1X20.1-1 IsBool[101X
  
  [33X[1;0Y[29X[2XIsBool[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0Ytests whether [3Xobj[103X is [9Xtrue[109X, [9Xfalse[109X or [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsBool( true );  IsBool( false );  IsBool( 17 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X20.2 [33X[0;0YFail (Variable)[133X[101X
  
  [1X20.2-1 fail[101X
  
  [33X[1;0Y[29X[2Xfail[102X [32X global variable[133X
  
  [33X[0;0YThe value [9Xfail[109X is used to indicate situations when an operation could not be
  performed  for  the  given  arguments, either because of shortcomings of the
  arguments or because of restrictions in the implementation or computability.
  So for example [2XPosition[102X ([14X21.16-1[114X) will return [9Xfail[109X if the point searched for
  is not in the list.[133X
  
  [33X[0;0Y[9Xfail[109X  is  simply  an  object  that is different from every other object than
  itself.[133X
  
  [33X[0;0YFor technical reasons, [9Xfail[109X is a boolean value. But note that [9Xfail[109X cannot be
  used to form boolean expressions with [9Xand[109X, [9Xor[109X, and [9Xnot[109X (see [14X20.4[114X below), and
  [9Xfail[109X cannot appear in boolean lists (see Chapter [14X22[114X).[133X
  
  
  [1X20.3 [33X[0;0YComparisons of Booleans[133X[101X
  
  
  [1X20.3-1 [33X[0;0YEquality and inequality of Booleans[133X[101X
  
  [33X[0;0Y[10X[3Xbool1[103X[10X = [3Xbool2[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xbool1[103X[10X <> [3Xbool2[103X[10X[110X[133X
  
  [33X[0;0YThe  equality  operator  [10X=[110X evaluates to [9Xtrue[109X if the two boolean values [3Xbool1[103X
  and [3Xbool2[103X are equal, i.e., both are [9Xtrue[109X or both are [9Xfalse[109X or both [9Xfail[109X, and
  [9Xfalse[109X  otherwise.  The  inequality  operator [10X<>[110X evaluates to [9Xtrue[109X if the two
  boolean  values  [3Xbool1[103X,  [3Xbool2[103X  are  different,  and  [9Xfalse[109X  otherwise. This
  operation  is  also  called  the  [13Xexclusive or[113X, because its value is [9Xtrue[109X if
  exactly one of [3Xbool1[103X or [3Xbool2[103X is [9Xtrue[109X.[133X
  
  [33X[0;0YYou  can  compare boolean values with objects of other types. Of course they
  are never equal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue = false;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xfalse = (true = fail);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xtrue <> 17;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X20.3-2 [33X[0;0YOrdering of Booleans[133X[101X
  
  [33X[0;0Y[3Xbool1[103X [10X<[110X [3Xbool2[103X[133X
  
  [33X[0;0YThe  ordering  of  boolean values is defined by [9Xtrue[109X [10X<[110X [9Xfalse[109X [10X<[110X [9Xfail[109X. For the
  comparison of booleans with other [5XGAP[105X objects, see Section [14X4.13[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue < false;  fail >= false;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X20.4 [33X[0;0YOperations for Booleans[133X[101X
  
  [33X[0;0YThe following boolean operations are only applicable to [9Xtrue[109X and [9Xfalse[109X.[133X
  
  
  [1X20.4-1 [33X[0;0YLogical disjunction[133X[101X
  
  [33X[0;0Y[3Xbool1[103X [9Xor[109X [3Xbool2[103X[133X
  
  [33X[0;0YThe logical operator [9Xor[109X evaluates to [9Xtrue[109X if at least one of the two boolean
  operands [3Xbool1[103X and [3Xbool2[103X is [9Xtrue[109X, and to [9Xfalse[109X otherwise.[133X
  
  [33X[0;0Y[9Xor[109X first evaluates [3Xbool1[103X. If the value is neither [9Xtrue[109X nor [9Xfalse[109X an error is
  signalled.  If  the  value  is [9Xtrue[109X, then [9Xor[109X returns [9Xtrue[109X [13Xwithout[113X evaluating
  [3Xbool2[103X.  If  the value is [9Xfalse[109X, then [9Xor[109X evaluates [3Xbool2[103X. Again, if the value
  is  neither  [9Xtrue[109X  nor [9Xfalse[109X an error is signalled. Otherwise [9Xor[109X returns the
  value of [3Xbool2[103X. This [13Xshort-circuited[113X evaluation is important if the value of
  [3Xbool1[103X  is  [9Xtrue[109X  and  evaluation  of  [3Xbool2[103X would take much time or cause an
  error.[133X
  
  [33X[0;0Y[9Xor[109X  is  associative,  i.e.,  it is allowed to write [3Xb1[103X [9Xor[109X [3Xb2[103X [9Xor[109X [3Xb3[103X, which is
  interpreted as ([3Xb1[103X [9Xor[109X [3Xb2[103X) [9Xor[109X [3Xb3[103X. [9Xor[109X has the lowest precedence of the logical
  operators.  All  logical operators have lower precedence than the comparison
  operators [10X=[110X, [10X<[110X, [9Xin[109X, etc.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue or false;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xfalse or false;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xi := -1;; l := [1,2,3];;[127X[104X
    [4X[25Xgap>[125X [27Xif i <= 0 or l[i] = false then     # this does not cause an error,[127X[104X
    [4X[25X>[125X [27X   Print("aha\n"); fi;                # because `l[i]' is not evaluated[127X[104X
    [4X[28Xaha[128X[104X
  [4X[32X[104X
  
  
  [1X20.4-2 [33X[0;0YLogical conjunction[133X[101X
  
  [33X[0;0Y[3Xbool1[103X [9Xand[109X [3Xbool2[103X[133X
  
  [33X[0;0Y[3Xfil1[103X [9Xand[109X [3Xfil2[103X[133X
  
  [33X[0;0YThe  logical  operator [9Xand[109X evaluates to [9Xtrue[109X if both boolean operands [3Xbool1[103X,
  [3Xbool2[103X are [9Xtrue[109X, and to [9Xfalse[109X otherwise.[133X
  
  [33X[0;0Y[9Xand[109X  first  evaluates [3Xbool1[103X. If the value is neither [9Xtrue[109X nor [9Xfalse[109X an error
  is  signalled.  If  the  value  is  [9Xfalse[109X,  then  [9Xand[109X  returns [9Xfalse[109X [13Xwithout[113X
  evaluating  [3Xbool2[103X. If the value is [9Xtrue[109X, then [9Xand[109X evaluates [3Xbool2[103X. Again, if
  the  value  is  neither  [9Xtrue[109X nor [9Xfalse[109X an error is signalled. Otherwise [9Xand[109X
  returns  the value of [3Xbool2[103X. This [13Xshort-circuited[113X evaluation is important if
  the  value of [3Xbool1[103X is [9Xfalse[109X and evaluation of [3Xbool2[103X would take much time or
  cause an error.[133X
  
  [33X[0;0Y[9Xand[109X  is associative, i.e., it is allowed to write [3Xb1[103X [9Xand[109X [3Xb2[103X [9Xand[109X [3Xb3[103X, which is
  interpreted  as  ([3Xb1[103X  [9Xand[109X  [3Xb2[103X)  [9Xand[109X  [3Xb3[103X.  [9Xand[109X has higher precedence than the
  logical  [9Xor[109X  operator,  but  lower  than the unary logical [9Xnot[109X operator. All
  logical  operators have lower precedence than the comparison operators [10X=[110X, [10X<[110X,
  [9Xin[109X, etc.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue and false;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xtrue and true;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xfalse and 17;  # does not cause error, because 17 is never looked at[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[9Xand[109X can also be applied to filters. It returns a filter that when applied to
  some argument [3Xx[103X, tests [3Xfil1[103X[22X(x)[122X [9Xand[109X [3Xfil2[103X[22X(x)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xandfilt:= IsPosRat and IsInt;;[127X[104X
    [4X[25Xgap>[125X [27Xandfilt( 17 );  andfilt( 1/2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X20.4-3 [33X[0;0YLogical negation[133X[101X
  
  [33X[0;0Y[9Xnot[109X [3Xbool[103X[133X
  
  [33X[0;0YThe  logical  operator  [9Xnot[109X returns [9Xtrue[109X if the boolean value [3Xbool[103X is [9Xfalse[109X,
  and  [9Xtrue[109X otherwise. An error is signalled if [3Xbool[103X does not evaluate to [9Xtrue[109X
  or [9Xfalse[109X.[133X
  
  [33X[0;0Y[9Xnot[109X  has higher precedence than the other logical operators, [9Xor[109X and [9Xand[109X. All
  logical  operators have lower precedence than the comparison operators [10X=[110X, [10X<[110X,
  [9Xin[109X, etc.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue and false;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xnot true;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xnot false;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
