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Maxima provides set functions, such as intersection and union, for finite sets that are defined by explicit enumeration. Maxima treats lists and sets as distinct objects. This feature makes it possible to work with sets that have members that are either lists or sets.
In addition to functions for finite sets, Maxima provides some functions related to combinatorics; these include the Stirling numbers of the first and second kind, the Bell numbers, multinomial coefficients, partitions of nonnegative integers, and a few others. Maxima also defines a Kronecker delta function.
To construct a set with members a_1, ..., a_n, write
set(a_1, ..., a_n) or {a_1, ..., a_n};
to construct the empty set, write set() or {}.
In input, set(...) and { ... } are equivalent.
Sets are always displayed with curly braces.
If a member is listed more than once, simplification eliminates the redundant member.
(%i1) set();
(%o1) {}
(%i2) set(a, b, a);
(%o2) {a, b}
(%i3) set(a, set(b));
(%o3) {a, {b}}
(%i4) set(a, [b]);
(%o4) {a, [b]}
(%i5) {};
(%o5) {}
(%i6) {a, b, a};
(%o6) {a, b}
(%i7) {a, {b}};
(%o7) {a, {b}}
(%i8) {a, [b]};
(%o8) {a, [b]}
Two would-be elements x and y are redundant
(i.e., considered the same for the purpose of set construction)
if and only if is(x = y) yields true.
Note that is(equal(x, y)) can yield true
while is(x = y) yields false;
in that case the elements x and y are considered distinct.
(%i1) x: a/c + b/c;
b a
(%o1) - + -
c c
(%i2) y: a/c + b/c;
b a
(%o2) - + -
c c
(%i3) z: (a + b)/c;
b + a
(%o3) -----
c
(%i4) is (x = y);
(%o4) true
(%i5) is (y = z);
(%o5) false
(%i6) is (equal (y, z));
(%o6) true
(%i7) y - z;
b + a b a
(%o7) - ----- + - + -
c c c
(%i8) ratsimp (%);
(%o8) 0
(%i9) {x, y, z};
b + a b a
(%o9) {-----, - + -}
c c c
To construct a set from the elements of a list, use setify.
(%i1) setify ([b, a]);
(%o1) {a, b}
Set members x and y are equal provided is(x = y)
evaluates to true. Thus rat(x) and x are equal as set
members; consequently,
(%i1) {x, rat(x)};
(%o1) {x}
Further, since is((x - 1)*(x + 1) = x^2 - 1) evaluates to false,
(x - 1)*(x + 1) and x^2 - 1 are distinct set members; thus
(%i1) {(x - 1)*(x + 1), x^2 - 1};
2
(%o1) {(x - 1) (x + 1), x - 1}
To reduce this set to a singleton set, apply rat to each set member:
(%i1) {(x - 1)*(x + 1), x^2 - 1};
2
(%o1) {(x - 1) (x + 1), x - 1}
(%i2) map (rat, %);
2
(%o2)/R/ {x - 1}
To remove redundancies from other sets, you may need to use other
simplification functions. Here is an example that uses trigsimp:
(%i1) {1, cos(x)^2 + sin(x)^2};
2 2
(%o1) {1, sin (x) + cos (x)}
(%i2) map (trigsimp, %);
(%o2) {1}
A set is simplified when its members are non-redundant and
sorted. The current version of the set functions uses the Maxima function
orderlessp to order sets; however, future versions of
the set functions might use a different ordering function.
Some operations on sets, such as substitution, automatically force a re-simplification; for example,
(%i1) s: {a, b, c}$
(%i2) subst (c=a, s);
(%o2) {a, b}
(%i3) subst ([a=x, b=x, c=x], s);
(%o3) {x}
(%i4) map (lambda ([x], x^2), set (-1, 0, 1));
(%o4) {0, 1}
Maxima treats lists and sets as distinct objects;
functions such as union and intersection complain
if any argument is not a set. If you need to apply a set
function to a list, use the setify function to convert it
to a set. Thus
(%i1) union ([1, 2], {a, b});
Function union expects a set, instead found [1,2]
-- an error. Quitting. To debug this try debugmode(true);
(%i2) union (setify ([1, 2]), {a, b});
(%o2) {1, 2, a, b}
To extract all set elements of a set s that satisfy a predicate
f, use subset(s, f). (A predicate is a
boolean-valued function.) For example, to find the equations
in a given set that do not depend on a variable z, use
(%i1) subset ({x + y + z, x - y + 4, x + y - 5},
lambda ([e], freeof (z, e)));
(%o1) {- y + x + 4, y + x - 5}
The section Functions and Variables for Sets has a complete list of the set functions in Maxima.
There two ways to to iterate over set members. One way is the use
map; for example:
(%i1) map (f, {a, b, c});
(%o1) {f(a), f(b), f(c)}
The other way is to use for x in s do
(%i1) s: {a, b, c};
(%o1) {a, b, c}
(%i2) for si in s do print (concat (si, 1));
a1
b1
c1
(%o2) done
The Maxima functions first and rest work
correctly on sets. Applied to a set, first returns the first
displayed element of a set; which element that is may be
implementation-dependent. If s is a set, then
rest(s) is equivalent to disjoin(first(s), s).
Currently, there are other Maxima functions that work correctly
on sets.
In future versions of the set functions,
first and rest may function differently or not at all.
Maxima’s orderless and ordergreat mechanisms are
incompatible with the set functions. If you need to use either orderless
or ordergreat, call those functions before constructing any sets,
and do not call unorder.
Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.
Previous: Introduction to Sets, Up: Sets [Contents][Index]
Returns the union of the set a with {x}.
adjoin complains if a is not a literal set.
adjoin(x, a) and union(set(x), a)
are equivalent;
however, adjoin may be somewhat faster than union.
See also disjoin.
Examples:
(%i1) adjoin (c, {a, b});
(%o1) {a, b, c}
(%i2) adjoin (a, {a, b});
(%o2) {a, b}
Represents the n-th Bell number.
belln(n) is the number of partitions of a set with n members.
For nonnegative integers n,
belln(n) simplifies to the n-th Bell number.
belln does not simplify for any other arguments.
belln distributes over equations, lists, matrices, and sets.
Examples:
belln applied to nonnegative integers.
(%i1) makelist (belln (i), i, 0, 6);
(%o1) [1, 1, 2, 5, 15, 52, 203]
(%i2) is (cardinality (set_partitions ({})) = belln (0));
(%o2) true
(%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
belln (6));
(%o3) true
belln applied to arguments which are not nonnegative integers.
(%i1) [belln (x), belln (sqrt(3)), belln (-9)]; (%o1) [belln(x), belln(sqrt(3)), belln(- 9)]
Returns the number of distinct elements of the set a.
cardinality ignores redundant elements
even when simplification is disabled.
Examples:
(%i1) cardinality ({});
(%o1) 0
(%i2) cardinality ({a, a, b, c});
(%o2) 3
(%i3) simp : false;
(%o3) false
(%i4) cardinality ({a, a, b, c});
(%o4) 3
Returns a set of lists of the form [x_1, ..., x_n], where
x_1, ..., x_n are elements of the sets b_1, ... , b_n,
respectively.
cartesian_product complains if any argument is not a literal set.
See also cartesian_product_list.
Examples:
(%i1) cartesian_product ({0, 1});
(%o1) {[0], [1]}
(%i2) cartesian_product ({0, 1}, {0, 1});
(%o2) {[0, 0], [0, 1], [1, 0], [1, 1]}
(%i3) cartesian_product ({x}, {y}, {z});
(%o3) {[x, y, z]}
(%i4) cartesian_product ({x}, {-1, 0, 1});
(%o4) {[x, - 1], [x, 0], [x, 1]}
Returns a list of lists of the form [x_1, ..., x_n], where
x_1, ..., x_n are elements of the lists b_1, ... , b_n, respectively,
comprising all possible combinations of the elements of b_1, ... , b_n.
The list returned by cartesian_product_list is equivalent to the
following recursive definition.
Let L be the list returned by cartesian_product_list(b_2, ..., b_n).
Then cartesian_product_list(b_1, b_2, ..., b_n)
(i.e., b_1 in addition to b_2, ..., b_n)
returns a list comprising each element of L appended to the first element of b_1,
each element of L appended to the second element of b_1,
each element of L appended to the third element of b_1, etc.
The order of the list returned by cartesian_product_list(b_1, b_2, ..., b_n)
may therefore be summarized by saying the lesser indices (1, 2, 3, ...) vary more slowly than the greater indices.
The list returned by cartesian_product_list contains duplicate elements
if any argument b_1, ..., b_n contains duplicates.
In this respect, cartesian_product_list differs from cartesian_product,
which returns no duplicates.
Also, the ordering of the list returned cartesian_product_list
is determined by the order of the elements of b_1, ..., b_n.
Again, this differs from cartesian_product,
which returns a set (with order determined by orderlessp).
The length of the list returned by cartesian_product_list
is equal to the product of the lengths of the arguments b_1, ..., b_n.
See also cartesian_product.
cartesian_product_list complains if any argument is not a list.
Examples:
cartesian_product_list returns a list of lists comprising all possible combinations.
(%i1) cartesian_product_list ([0, 1]);
(%o1) [[0], [1]]
(%i2) cartesian_product_list ([0, 1], [0, 1]);
(%o2) [[0, 0], [0, 1], [1, 0], [1, 1]]
(%i3) cartesian_product_list ([x], [y], [z]);
(%o3) [[x, y, z]]
(%i4) cartesian_product_list ([x], [-1, 0, 1]);
(%o4) [[x, - 1], [x, 0], [x, 1]]
(%i5) cartesian_product_list ([a, h, e], [c, b, 4]);
(%o5) [[a, c], [a, b], [a, 4], [h, c], [h, b], [h, 4], [e, c],
[e, b], [e, 4]]
The order of the list returned by cartesian_product_list
may be summarized by saying the lesser indices vary more slowly than the greater indices.
(%i1) cartesian_product_list ([1, 2, 3], [a, b], [i, ii]); (%o1) [[1, a, i], [1, a, ii], [1, b, i], [1, b, ii], [2, a, i], [2, a, ii], [2, b, i], [2, b, ii], [3, a, i], [3, a, ii], [3, b, i], [3, b, ii]]
The list returned by cartesian_product_list contains duplicate elements
if any argument contains duplicates.
(%i1) cartesian_product_list ([e, h], [3, 7, 3]); (%o1) [[e, 3], [e, 7], [e, 3], [h, 3], [h, 7], [h, 3]]
The length of the list returned by cartesian_product_list
is equal to the product of the lengths of the arguments.
(%i1) foo: cartesian_product_list ([1, 1, 2, 2, 3], [h, z, h]); (%o1) [[1, h], [1, z], [1, h], [1, h], [1, z], [1, h], [2, h], [2, z], [2, h], [2, h], [2, z], [2, h], [3, h], [3, z], [3, h]] (%i2) is (length (foo) = 5*3); (%o2) true
Returns the set a without the member x. If x is not a member of a, return a unchanged.
disjoin complains if a is not a literal set.
disjoin(x, a), delete(x, a), and
setdifference(a, set(x)) are all equivalent.
Of these, disjoin is generally faster than the others.
Examples:
(%i1) disjoin (a, {a, b, c, d});
(%o1) {b, c, d}
(%i2) disjoin (a + b, {5, z, a + b, %pi});
(%o2) {5, %pi, z}
(%i3) disjoin (a - b, {5, z, a + b, %pi});
(%o3) {5, %pi, b + a, z}
Returns true if and only if the sets a and b are disjoint.
disjointp complains if either a or b is not a literal set.
Examples:
(%i1) disjointp ({a, b, c}, {1, 2, 3});
(%o1) true
(%i2) disjointp ({a, b, 3}, {1, 2, 3});
(%o2) false
Represents the set of divisors of n.
divisors(n) simplifies to a set of integers
when n is a nonzero integer.
The set of divisors includes the members 1 and n.
The divisors of a negative integer are the divisors of its absolute value.
divisors distributes over equations, lists, matrices, and sets.
Examples:
We can verify that 28 is a perfect number: the sum of its divisors (except for itself) is 28.
(%i1) s: divisors(28);
(%o1) {1, 2, 4, 7, 14, 28}
(%i2) lreduce ("+", args(s)) - 28;
(%o2) 28
divisors is a simplifying function.
Substituting 8 for a in divisors(a)
yields the divisors without reevaluating divisors(8).
(%i1) divisors (a);
(%o1) divisors(a)
(%i2) subst (8, a, %);
(%o2) {1, 2, 4, 8}
divisors distributes over equations, lists, matrices, and sets.
(%i1) divisors (a = b);
(%o1) divisors(a) = divisors(b)
(%i2) divisors ([a, b, c]);
(%o2) [divisors(a), divisors(b), divisors(c)]
(%i3) divisors (matrix ([a, b], [c, d]));
[ divisors(a) divisors(b) ]
(%o3) [ ]
[ divisors(c) divisors(d) ]
(%i4) divisors ({a, b, c});
(%o4) {divisors(a), divisors(b), divisors(c)}
Returns true if and only if x is a member of the
set a.
elementp complains if a is not a literal set.
Examples:
(%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
(%o1) true
(%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
(%o2) false
Return true if and only if a is the empty set or
the empty list.
Examples:
(%i1) map (emptyp, [{}, []]);
(%o1) [true, true]
(%i2) map (emptyp, [a + b, {{}}, %pi]);
(%o2) [false, false, false]
Returns a set of the equivalence classes of the set s with respect to the equivalence relation F.
F is a function of two variables defined on the Cartesian product of s with s.
The return value of F is either true or false,
or an expression expr such that is(expr) is either true or false.
When F is not an equivalence relation,
equiv_classes accepts it without complaint,
but the result is generally incorrect in that case.
Examples:
The equivalence relation is a lambda expression which returns true or false.
(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
lambda ([x, y], is (equal (x, y))));
(%o1) {{1, 1.0}, {2, 2.0}, {3, 3.0}}
The equivalence relation is the name of a relational function
which is evaluates to true or false.
(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
(%o1) {{1, 1.0}, {2, 2.0}, {3, 3.0}}
The equivalence classes are numbers which differ by a multiple of 3.
(%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
lambda ([x, y], remainder (x - y, 3) = 0));
(%o1) {{1, 4, 7}, {2, 5}, {3, 6}}
Returns true if the predicate f is true for all given arguments.
Given one set as the second argument,
every(f, s) returns true
if is(f(a_i)) returns true for all a_i in s.
every may or may not evaluate f for all a_i in s.
Since sets are unordered,
every may evaluate f(a_i) in any order.
Given one or more lists as arguments,
every(f, L_1, ..., L_n) returns true
if is(f(x_1, ..., x_n)) returns true
for all x_1, ..., x_n in L_1, ..., L_n, respectively.
every may or may not evaluate
f for every combination x_1, ..., x_n.
every evaluates lists in the order of increasing index.
Given an empty set {} or empty lists [] as arguments,
every returns true.
When the global flag maperror is true, all lists
L_1, ..., L_n must have equal lengths.
When maperror is false, list arguments are
effectively truncated to the length of the shortest list.
Return values of the predicate f which evaluate (via is)
to something other than true or false
are governed by the global flag prederror.
When prederror is true,
such values are treated as false,
and the return value from every is false.
When prederror is false,
such values are treated as unknown,
and the return value from every is unknown.
Examples:
every applied to a single set.
The predicate is a function of one argument.
(%i1) every (integerp, {1, 2, 3, 4, 5, 6});
(%o1) true
(%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2) false
every applied to two lists.
The predicate is a function of two arguments.
(%i1) every ("=", [a, b, c], [a, b, c]);
(%o1) true
(%i2) every ("#", [a, b, c], [a, b, c]);
(%o2) false
Return values of the predicate f which evaluate
to something other than true or false
are governed by the global flag prederror.
(%i1) prederror : false;
(%o1) false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
[x^2, y^2, z^2]);
(%o2) [unknown, unknown, unknown]
(%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3) unknown
(%i4) prederror : true;
(%o4) true
(%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o5) false
Returns the subset of s for which the function f takes on maximum or minimum values.
extremal_subset(s, f, max) returns the subset of the set or
list s for which the real-valued function f takes on its maximum value.
extremal_subset(s, f, min) returns the subset of the set or
list s for which the real-valued function f takes on its minimum value.
Examples:
(%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
(%o1) {- 2, 2}
(%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
(%o2) {sqrt(2)}
Collects arguments of subexpressions which have the same operator as expr and constructs an expression from these collected arguments.
Subexpressions in which the operator is different from the main operator of expr
are copied without modification,
even if they, in turn, contain some subexpressions in which the operator is the same as for expr.
It may be possible for flatten to construct expressions in which the number
of arguments differs from the declared arguments for an operator;
this may provoke an error message from the simplifier or evaluator.
flatten does not try to detect such situations.
Expressions with special representations, for example, canonical rational expressions (CRE),
cannot be flattened; in such cases, flatten returns its argument unchanged.
Examples:
Applied to a list, flatten gathers all list elements that are lists.
(%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]); (%o1) [a, b, c, d, e, f, g, h, i, j]
Applied to a set, flatten gathers all members of set elements that are sets.
(%i1) flatten ({a, {b}, {{c}}});
(%o1) {a, b, c}
(%i2) flatten ({a, {[a], {a}}});
(%o2) {a, [a]}
flatten is similar to the effect of declaring the main operator n-ary.
However, flatten has no effect on subexpressions which have an operator
different from the main operator, while an n-ary declaration affects those.
(%i1) expr: flatten (f (g (f (f (x))))); (%o1) f(g(f(f(x)))) (%i2) declare (f, nary); (%o2) done (%i3) ev (expr); (%o3) f(g(f(x)))
flatten treats subscripted functions the same as any other operator.
(%i1) flatten (f[5] (f[5] (x, y), z));
(%o1) f (x, y, z)
5
It may be possible for flatten to construct expressions in which the number
of arguments differs from the declared arguments for an operator;
(%i1) 'mod (5, 'mod (7, 4)); (%o1) mod(5, mod(7, 4)) (%i2) flatten (%); (%o2) mod(5, 7, 4) (%i3) ''%, nouns; Wrong number of arguments to mod -- an error. Quitting. To debug this try debugmode(true);
Replaces every set operator in a by a list operator,
and returns the result.
full_listify replaces set operators in nested subexpressions,
even if the main operator is not set.
listify replaces only the main operator.
Examples:
(%i1) full_listify ({a, b, {c, {d, e, f}, g}});
(%o1) [a, b, [c, [d, e, f], g]]
(%i2) full_listify (F (G ({a, b, H({c, d, e})})));
(%o2) F(G([a, b, H([c, d, e])]))
When a is a list, replaces the list operator with a set operator,
and applies fullsetify to each member which is a set.
When a is not a list, it is returned unchanged.
setify replaces only the main operator.
Examples:
In line (%o2), the argument of f isn’t converted to a set
because the main operator of f([b]) isn’t a list.
(%i1) fullsetify ([a, [a]]);
(%o1) {a, {a}}
(%i2) fullsetify ([a, f([b])]);
(%o2) {a, f([b])}
Returns x for any argument x.
Examples:
identity may be used as a predicate when the arguments
are already Boolean values.
(%i1) every (identity, [true, true]); (%o1) true
Returns integer partitions of n, that is, lists of integers which sum to n.
integer_partitions(n) returns the set of
all partitions of the integer n.
Each partition is a list sorted from greatest to least.
integer_partitions(n, len)
returns all partitions that have length len or less; in this
case, zeros are appended to each partition with fewer than len
terms to make each partition have exactly len terms.
Each partition is a list sorted from greatest to least.
A list [a_1, ..., a_m] is a partition of a nonnegative integer n when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.
Examples:
(%i1) integer_partitions (3);
(%o1) {[1, 1, 1], [2, 1], [3]}
(%i2) s: integer_partitions (25)$
(%i3) cardinality (s);
(%o3) 1958
(%i4) map (lambda ([x], apply ("+", x)), s);
(%o4) {25}
(%i5) integer_partitions (5, 3);
(%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
(%i6) integer_partitions (5, 2);
(%o6) {[3, 2], [4, 1], [5, 0]}
To find all partitions that satisfy a condition, use the function subset;
here is an example that finds all partitions of 10 that consist of prime numbers.
(%i1) s: integer_partitions (10)$
(%i2) cardinality (s);
(%o2) 42
(%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
(%i4) subset (s, lambda ([x], every (xprimep, x)));
(%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}
intersect is the same as intersection, which see.
Returns a set containing the elements that are common to the sets a_1 through a_n.
intersection complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c, d};
(%o1) {a, b, c, d}
(%i2) S_2 : {d, e, f, g};
(%o2) {d, e, f, g}
(%i3) S_3 : {c, d, e, f};
(%o3) {c, d, e, f}
(%i4) S_4 : {u, v, w};
(%o4) {u, v, w}
(%i5) intersection (S_1, S_2);
(%o5) {d}
(%i6) intersection (S_2, S_3);
(%o6) {d, e, f}
(%i7) intersection (S_1, S_2, S_3);
(%o7) {d}
(%i8) intersection (S_1, S_2, S_3, S_4);
(%o8) {}
Represents the Kronecker delta function.
kron_delta simplifies to 1 when xi and yj are equal
for all pairs of arguments, and it simplifies to 0 when xi and
yj are not equal for some pair of arguments. Equality is
determined using is(equal(xi,xj)) and inequality by
is(notequal(xi,xj)). For exactly one argument, kron_delta
signals an error.
Examples:
(%i1) kron_delta(a,a); (%o1) 1 (%i2) kron_delta(a,b,a,b); (%o2) kron_delta(a, b) (%i3) kron_delta(a,a,b,a+1); (%o3) 0 (%i4) assume(equal(x,y)); (%o4) [equal(x, y)] (%i5) kron_delta(x,y); (%o5) 1
Returns a list containing the members of a when a is a set.
Otherwise, listify returns a.
full_listify replaces all set operators in a by list operators.
Examples:
(%i1) listify ({a, b, c, d});
(%o1) [a, b, c, d]
(%i2) listify (F ({a, b, c, d}));
(%o2) F({a, b, c, d})
Returns a set with members generated from the expression expr, where x is a list of variables in expr, and s is a set or list of lists. To generate each set member, expr is evaluated with the variables x bound in parallel to a member of s.
Each member of s must have the same length as x. The list of variables x must be a list of symbols, without subscripts. Even if there is only one symbol, x must be a list of one element, and each member of s must be a list of one element.
See also makelist.
Examples:
(%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
1 2 3 4
(%o1) {-, -, -, -}
a b c d
(%i2) S : {x, y, z}$
(%i3) S3 : cartesian_product (S, S, S);
(%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y],
[x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x],
[y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z],
[y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y],
[z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x],
[z, z, y], [z, z, z]}
(%i4) makeset (i + j + k, [i, j, k], S3);
(%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x,
z + 2 y, 2 z + x, 2 z + y}
(%i5) makeset (sin(x), [x], {[1], [2], [3]});
(%o5) {sin(1), sin(2), sin(3)}
Represents the Moebius function.
When n is product of k distinct primes,
moebius(n) simplifies to (-1)^k;
when n = 1, it simplifies to 1;
and it simplifies to 0 for all other positive integers.
moebius distributes over equations, lists, matrices, and sets.
Examples:
(%i1) moebius (1);
(%o1) 1
(%i2) moebius (2 * 3 * 5);
(%o2) - 1
(%i3) moebius (11 * 17 * 29 * 31);
(%o3) 1
(%i4) moebius (2^32);
(%o4) 0
(%i5) moebius (n);
(%o5) moebius(n)
(%i6) moebius (n = 12);
(%o6) moebius(n) = 0
(%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
(%o7) [- 1, 1, 1]
(%i8) moebius (matrix ([11, 12], [13, 14]));
[ - 1 0 ]
(%o8) [ ]
[ - 1 1 ]
(%i9) moebius ({21, 22, 23, 24});
(%o9) {- 1, 0, 1}
Returns the multinomial coefficient.
When each a_k is a nonnegative integer, the multinomial coefficient
gives the number of ways of placing a_1 + ... + a_n
distinct objects into n boxes with a_k elements in the
k’th box. In general, multinomial_coeff (a_1, ..., a_n)
evaluates to (a_1 + ... + a_n)!/(a_1! ... a_n!).
multinomial_coeff() (with no arguments) evaluates to 1.
minfactorial may be able to simplify the value returned by multinomial_coeff.
Examples:
(%i1) multinomial_coeff (1, 2, x);
(x + 3)!
(%o1) --------
2 x!
(%i2) minfactorial (%);
(x + 1) (x + 2) (x + 3)
(%o2) -----------------------
2
(%i3) multinomial_coeff (-6, 2);
(- 4)!
(%o3) --------
2 (- 6)!
(%i4) minfactorial (%);
(%o4) 10
Returns the number of distinct integer partitions of n
when n is a nonnegative integer.
Otherwise, num_distinct_partitions returns a noun expression.
num_distinct_partitions(n, list) returns a
list of the number of distinct partitions of 1, 2, 3, ..., n.
A distinct partition of n is a list of distinct positive integers k_1, ..., k_m such that n = k_1 + ... + k_m.
Examples:
(%i1) num_distinct_partitions (12); (%o1) 15 (%i2) num_distinct_partitions (12, list); (%o2) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15] (%i3) num_distinct_partitions (n); (%o3) num_distinct_partitions(n)
Returns the number of integer partitions of n
when n is a nonnegative integer.
Otherwise, num_partitions returns a noun expression.
num_partitions(n, list) returns a
list of the number of integer partitions of 1, 2, 3, ..., n.
For a nonnegative integer n, num_partitions(n) is equal to
cardinality(integer_partitions(n)); however, num_partitions
does not actually construct the set of partitions, so it is much faster.
Examples:
(%i1) num_partitions (5) = cardinality (integer_partitions (5)); (%o1) 7 = 7 (%i2) num_partitions (8, list); (%o2) [1, 1, 2, 3, 5, 7, 11, 15, 22] (%i3) num_partitions (n); (%o3) num_partitions(n)
Partitions the set a according to the predicate f.
partition_set returns a list of two sets.
The first set comprises the elements of a for which f evaluates to false,
and the second comprises any other elements of a.
partition_set does not apply is to the return value of f.
partition_set complains if a is not a literal set.
See also subset.
Examples:
(%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
(%o1) [{1, 7}, {2, 8}]
(%i2) partition_set ({x, rat(y), rat(y) + z, 1},
lambda ([x], ratp(x)));
(%o2)/R/ [{1, x}, {y, y + z}]
Returns a set of all distinct permutations of the members of the list or set a. Each permutation is a list, not a set.
When a is a list, duplicate members of a are included in the permutations.
permutations complains if a is not a literal list or set.
See also random_permutation.
Examples:
(%i1) permutations ([a, a]);
(%o1) {[a, a]}
(%i2) permutations ([a, a, b]);
(%o2) {[a, a, b], [a, b, a], [b, a, a]}
Returns the set of all subsets of a, or a subset of that set.
powerset(a) returns the set of all subsets of the set a.
powerset(a) has 2^cardinality(a) members.
powerset(a, n) returns the set of all subsets of a that have
cardinality n.
powerset complains if a is not a literal set,
or if n is not a nonnegative integer.
Examples:
(%i1) powerset ({a, b, c});
(%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
(%i2) powerset ({w, x, y, z}, 4);
(%o2) {{w, x, y, z}}
(%i3) powerset ({w, x, y, z}, 3);
(%o3) {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
(%i4) powerset ({w, x, y, z}, 2);
(%o4) {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
(%i5) powerset ({w, x, y, z}, 1);
(%o5) {{w}, {x}, {y}, {z}}
(%i6) powerset ({w, x, y, z}, 0);
(%o6) {{}}
Returns a random permutation of the set or list a, as constructed by the Knuth shuffle algorithm.
The return value is a new list, which is distinct from the argument even if all elements happen to be the same. However, the elements of the argument are not copied.
Examples:
(%i1) random_permutation ([a, b, c, 1, 2, 3]);
(%o1) [c, 1, 2, 3, a, b]
(%i2) random_permutation ([a, b, c, 1, 2, 3]);
(%o2) [b, 3, 1, c, a, 2]
(%i3) random_permutation ({x + 1, y + 2, z + 3});
(%o3) [y + 2, z + 3, x + 1]
(%i4) random_permutation ({x + 1, y + 2, z + 3});
(%o4) [x + 1, y + 2, z + 3]
Returns a set containing the elements in the set a that are not in the set b.
setdifference complains if either a or b is not a literal set.
Examples:
(%i1) S_1 : {a, b, c, x, y, z};
(%o1) {a, b, c, x, y, z}
(%i2) S_2 : {aa, bb, c, x, y, zz};
(%o2) {aa, bb, c, x, y, zz}
(%i3) setdifference (S_1, S_2);
(%o3) {a, b, z}
(%i4) setdifference (S_2, S_1);
(%o4) {aa, bb, zz}
(%i5) setdifference (S_1, S_1);
(%o5) {}
(%i6) setdifference (S_1, {});
(%o6) {a, b, c, x, y, z}
(%i7) setdifference ({}, S_1);
(%o7) {}
Returns true if sets a and b have the same number of elements
and is(x = y) is true
for x in the elements of a
and y in the elements of b,
considered in the order determined by listify.
Otherwise, setequalp returns false.
Examples:
(%i1) setequalp ({1, 2, 3}, {1, 2, 3});
(%o1) true
(%i2) setequalp ({a, b, c}, {1, 2, 3});
(%o2) false
(%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
(%o3) false
Constructs a set from the elements of the list a. Duplicate
elements of the list a are deleted and the elements
are sorted according to the predicate orderlessp.
setify complains if a is not a literal list.
Examples:
(%i1) setify ([1, 2, 3, a, b, c]);
(%o1) {1, 2, 3, a, b, c}
(%i2) setify ([a, b, c, a, b, c]);
(%o2) {a, b, c}
(%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
(%o3) {1, 3, 5, 7, 9, 11, 13}
Returns true if and only if a is a Maxima set.
setp returns true for unsimplified sets (that is, sets with redundant members)
as well as simplified sets.
setp is equivalent to the Maxima function
setp(a) := not atom(a) and op(a) = 'set.
Examples:
(%i1) simp : false;
(%o1) false
(%i2) {a, a, a};
(%o2) {a, a, a}
(%i3) setp (%);
(%o3) true
Returns the set of all partitions of a, or a subset of that set.
set_partitions(a, n) returns a set of all
decompositions of a into n nonempty disjoint subsets.
set_partitions(a) returns the set of all partitions.
stirling2 returns the cardinality of the set of partitions of a set.
A set of sets P is a partition of a set S when
Examples:
The empty set is a partition of itself, the conditions 1 and 2 being vacuously true.
(%i1) set_partitions ({});
(%o1) {{}}
The cardinality of the set of partitions of a set can be found using stirling2.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) cardinality(p) = stirling2 (6, 3);
(%o3) 90 = 90
Each member of p should have n = 3 members; let’s check.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) map (cardinality, p);
(%o3) {3}
Finally, for each member of p, the union of its members should
equal s; again let’s check.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) map (lambda ([x], apply (union, listify (x))), p);
(%o3) {{0, 1, 2, 3, 4, 5}}
Returns true if the predicate f is true for one or more given arguments.
Given one set as the second argument,
some(f, s) returns true
if is(f(a_i)) returns true for one or more a_i in s.
some may or may not evaluate f for all a_i in s.
Since sets are unordered,
some may evaluate f(a_i) in any order.
Given one or more lists as arguments,
some(f, L_1, ..., L_n) returns true
if is(f(x_1, ..., x_n)) returns true
for one or more x_1, ..., x_n in L_1, ..., L_n, respectively.
some may or may not evaluate
f for some combinations x_1, ..., x_n.
some evaluates lists in the order of increasing index.
Given an empty set {} or empty lists [] as arguments,
some returns false.
When the global flag maperror is true, all lists
L_1, ..., L_n must have equal lengths.
When maperror is false, list arguments are
effectively truncated to the length of the shortest list.
Return values of the predicate f which evaluate (via is)
to something other than true or false
are governed by the global flag prederror.
When prederror is true,
such values are treated as false.
When prederror is false,
such values are treated as unknown.
Examples:
some applied to a single set.
The predicate is a function of one argument.
(%i1) some (integerp, {1, 2, 3, 4, 5, 6});
(%o1) true
(%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2) true
some applied to two lists.
The predicate is a function of two arguments.
(%i1) some ("=", [a, b, c], [a, b, c]);
(%o1) true
(%i2) some ("#", [a, b, c], [a, b, c]);
(%o2) false
Return values of the predicate f which evaluate
to something other than true or false
are governed by the global flag prederror.
(%i1) prederror : false;
(%o1) false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
[x^2, y^2, z^2]);
(%o2) [unknown, unknown, unknown]
(%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3) unknown
(%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o4) true
(%i5) prederror : true;
(%o5) true
(%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o6) false
(%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o7) true
Represents the Stirling number of the first kind.
When n and m are nonnegative
integers, the magnitude of stirling1 (n, m) is the number of
permutations of a set with n members that have m cycles.
stirling1 is a simplifying function.
Maxima knows the following identities:
These identities are applied when the arguments are literal integers
or symbols declared as integers, and the first argument is nonnegative.
stirling1 does not simplify for non-integer arguments.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n, n); (%o3) 1
Represents the Stirling number of the second kind.
When n and m are nonnegative
integers, stirling2 (n, m) is the number of ways a set with
cardinality n can be partitioned into m disjoint subsets.
stirling2 is a simplifying function.
Maxima knows the following identities.
These identities are applied when the arguments are literal integers
or symbols declared as integers, and the first argument is nonnegative.
stirling2 does not simplify for non-integer arguments.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling2 (n, n); (%o3) 1
stirling2 does not simplify for non-integer arguments.
(%i1) stirling2 (%pi, %pi); (%o1) stirling2(%pi, %pi)
Returns the subset of the set a that satisfies the predicate f.
subset returns a set which comprises the elements of a
for which f returns anything other than false.
subset does not apply is to the return value of f.
subset complains if a is not a literal set.
See also partition_set.
Examples:
(%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
(%o1) {1, 2, x, z}
(%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
(%o2) {2, 8, 14}
Returns true if and only if the set a is a subset of b.
subsetp complains if either a or b is not a literal set.
Examples:
(%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
(%o1) true
(%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
(%o2) false
Returns the symmetric difference of sets a_1, …, a_n.
Given two arguments, symmdifference (a, b) is the same as
union (setdifference (a, b), setdifference (b,
a)).
symmdifference complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c};
(%o1) {a, b, c}
(%i2) S_2 : {1, b, c};
(%o2) {1, b, c}
(%i3) S_3 : {a, b, z};
(%o3) {a, b, z}
(%i4) symmdifference ();
(%o4) {}
(%i5) symmdifference (S_1);
(%o5) {a, b, c}
(%i6) symmdifference (S_1, S_2);
(%o6) {1, a}
(%i7) symmdifference (S_1, S_2, S_3);
(%o7) {1, b, z}
(%i8) symmdifference ({}, S_1, S_2, S_3);
(%o8) {1,b, z}
Returns the union of the sets a_1 through a_n.
union() (with no arguments) returns the empty set.
union complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c + d, %e};
(%o1) {%e, a, b, d + c}
(%i2) S_2 : {%pi, %i, %e, c + d};
(%o2) {%e, %i, %pi, d + c}
(%i3) S_3 : {17, 29, 1729, %pi, %i};
(%o3) {17, 29, 1729, %i, %pi}
(%i4) union ();
(%o4) {}
(%i5) union (S_1);
(%o5) {%e, a, b, d + c}
(%i6) union (S_1, S_2);
(%o6) {%e, %i, %pi, a, b, d + c}
(%i7) union (S_1, S_2, S_3);
(%o7) {17, 29, 1729, %e, %i, %pi, a, b, d + c}
(%i8) union ({}, S_1, S_2, S_3);
(%o8) {17, 29, 1729, %e, %i, %pi, a, b, d + c}
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