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35.1 Introduction to Sets | ||
35.2 Functions and Variables for Sets |
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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.
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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.
Categories: Sets
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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.
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The set functions use the Maxima function orderlessp
to
order set members and the (Lisp-level) function like
to test for set
member equality. Both of these functions have known bugs
that may manifest if you attempt to use
sets with members that are lists or matrices that contain expressions
in canonical rational expression (CRE) form. An example is
(%i1) {[x], [rat (x)]}; Maxima encountered a Lisp error: The value #:X1440 is not of type LIST. Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil.
This expression causes Maxima to halt with an error (the error message depends on which version of Lisp your Maxima uses). Another example is
(%i1) setify ([[rat(a)], [rat(b)]]); Maxima encountered a Lisp error: The value #:A1440 is not of type LIST. Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil.
These bugs are caused by bugs in orderlessp
and like
; they
are not caused by bugs in the set functions. To illustrate, try the expressions
(%i1) orderlessp ([rat(a)], [rat(b)]); Maxima encountered a Lisp error: The value #:B1441 is not of type LIST. Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. (%i2) is ([rat(a)] = [rat(a)]); (%o2) false
Until these bugs are fixed, do not construct sets with members that are lists or matrices containing expressions in CRE form; a set with a member in CRE form, however, shouldn't be a problem:
(%i1) {x, rat (x)}; (%o1) {x}
Maxima's orderlessp
has another bug that can cause problems
with set functions, namely that the ordering predicate orderlessp
is
not transitive. The simplest known example that shows this is
(%i1) q: x^2$ (%i2) r: (x + 1)^2$ (%i3) s: x*(x + 2)$ (%i4) orderlessp (q, r); (%o4) true (%i5) orderlessp (r, s); (%o5) true (%i6) orderlessp (q, s); (%o6) false
This bug can cause trouble with all set functions as well as with
Maxima functions in general. It is probable, but not certain, that
this bug can be avoided
if all set members are either in CRE form or have been simplified
using ratsimp
.
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
.
If you find something that you think might be a set function bug, please
report it to the Maxima bug database. See bug_report
.
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Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.
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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}
Categories: Sets
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)]
Categories: Sets
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
Categories: Sets
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.
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]}
Categories: Sets
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}
Categories: Sets
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
Categories: Sets Predicate functions
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)}
Categories: Integers
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
Categories: Sets Predicate functions
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]
Categories: Sets Predicate functions
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}}
Categories: Sets
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 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
,
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
Categories: Sets
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)}
Categories: Sets
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])]))
Categories: Sets
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])}
Categories: Lists
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]}
Categories: Integers
intersect
is the same as intersection
, which see.
Categories: Sets
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) {}
Categories: Sets
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})
Categories: Sets
Extends the binary function F to an n-ary function by composition, where s is a list.
lreduce(F, s)
returns F(... F(F(s_1, s_2), s_3), ... s_n)
.
When the optional argument s_0 is present,
the result is equivalent to lreduce(F, cons(s_0, s))
.
The function F is first applied to the leftmost list elements, thus the name "lreduce".
See also rreduce
, xreduce
, and tree_reduce
.
Examples:
lreduce
without the optional argument.
(%i1) lreduce (f, [1, 2, 3]); (%o1) f(f(1, 2), 3) (%i2) lreduce (f, [1, 2, 3, 4]); (%o2) f(f(f(1, 2), 3), 4)
lreduce
with the optional argument.
(%i1) lreduce (f, [1, 2, 3], 4); (%o1) f(f(f(4, 1), 2), 3)
lreduce
applied to built-in binary operators.
/
is the division operator.
(%i1) lreduce ("^", args ({a, b, c, d})); b c d (%o1) ((a ) ) (%i2) lreduce ("/", args ({a, b, c, d})); a (%o2) ----- b c d
Categories: Lists
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)}
Categories: Sets
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}
Categories: Integers
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
Categories: Integers
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)
Categories: Integers
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)
Categories: Integers
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}]
Categories: Sets
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) {{}}
Categories: Sets
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]
Extends the binary function F to an n-ary function by composition, where s is a list.
rreduce(F, s)
returns F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n)))
.
When the optional argument s_{n + 1} is present,
the result is equivalent to rreduce(F, endcons(s_{n + 1}, s))
.
The function F is first applied to the rightmost list elements, thus the name "rreduce".
See also lreduce
, tree_reduce
, and xreduce
.
Examples:
rreduce
without the optional argument.
(%i1) rreduce (f, [1, 2, 3]); (%o1) f(1, f(2, 3)) (%i2) rreduce (f, [1, 2, 3, 4]); (%o2) f(1, f(2, f(3, 4)))
rreduce
with the optional argument.
(%i1) rreduce (f, [1, 2, 3], 4); (%o1) f(1, f(2, f(3, 4)))
rreduce
applied to built-in binary operators.
/
is the division operator.
(%i1) rreduce ("^", args ({a, b, c, d})); d c b (%o1) a (%i2) rreduce ("/", args ({a, b, c, d})); a c (%o2) --- b d
Categories: Lists
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) {}
Categories: Sets
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
Categories: Sets Predicate functions
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}
Categories: Lists
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
Categories: Sets Predicate functions
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}}
Categories: Sets
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.
For details, see Graham, Knuth and Patashnik Concrete Mathematics.
Maxima uses a recursion relation to define stirling1 (n, m)
for
m less than 0; it is undefined for n less than 0 and for non-integer
arguments.
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.
References:
[1] Donald Knuth, The Art of Computer Programming, third edition, Volume 1, Section 1.2.6, Equations 48, 49, and 50.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n, n); (%o3) 1
stirling1
does not simplify for non-integer arguments.
(%i1) stirling1 (sqrt(2), sqrt(2)); (%o1) stirling1(sqrt(2), sqrt(2))
Maxima applies identities to stirling1
.
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n + 1, n); n (n + 1) (%o3) --------- 2 (%i4) stirling1 (n + 1, 1); (%o4) n!
Categories: Integers
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.
Maxima uses a recursion relation to define stirling2 (n, m)
for
m less than 0; it is undefined for n less than 0 and for non-integer
arguments.
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.
References:
[1] Donald Knuth. The Art of Computer Programming, third edition, Volume 1, Section 1.2.6, Equations 48, 49, and 50.
[2] Graham, Knuth, and Patashnik. Concrete Mathematics, Table 264.
[3] Abramowitz and Stegun. Handbook of Mathematical Functions, Section 24.1.4.
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)
Maxima applies identities to stirling2
.
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling2 (n + 9, n + 8); (n + 8) (n + 9) (%o3) --------------- 2 (%i4) stirling2 (n + 1, 2); n (%o4) 2 - 1
Categories: Integers
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}
Categories: Sets
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
Categories: Sets Predicate functions
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}
Categories: Sets
Extends the binary function F to an n-ary function by composition, where s is a set or list.
tree_reduce
is equivalent to the following:
Apply F to successive pairs of elements
to form a new list [F(s_1, s_2), F(s_3, s_4), ...]
,
carrying the final element unchanged if there are an odd number of elements.
Then repeat until the list is reduced to a single element, which is the return value.
When the optional argument s_0 is present,
the result is equivalent tree_reduce(F, cons(s_0, s))
.
For addition of floating point numbers,
tree_reduce
may return a sum that has a smaller rounding error
than either rreduce
or lreduce
.
The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".
Examples:
tree_reduce
applied to a list with an even number of elements.
(%i1) tree_reduce (f, [a, b, c, d]); (%o1) f(f(a, b), f(c, d))
tree_reduce
applied to a list with an odd number of elements.
(%i1) tree_reduce (f, [a, b, c, d, e]); (%o1) f(f(f(a, b), f(c, d)), e)
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}
Categories: Sets
Extends the function F to an n-ary function by composition,
or, if F is already n-ary, applies F to s.
When F is not n-ary, xreduce
is the same as lreduce
.
The argument s is a list.
Functions known to be n-ary include
addition +
, multiplication *
, and
, or
, max
,
min
, and append
.
Functions may also be declared n-ary by declare(F, nary)
.
For these functions,
xreduce
is expected to be faster than either rreduce
or lreduce
.
When the optional argument s_0 is present,
the result is equivalent to xreduce(s, cons(s_0, s))
.
Floating point addition is not exactly associative; be that as it may,
xreduce
applies Maxima's n-ary addition when s contains floating point numbers.
Examples:
xreduce
applied to a function known to be n-ary.
F
is called once, with all arguments.
(%i1) declare (F, nary); (%o1) done (%i2) F ([L]) := L; (%o2) F([L]) := L (%i3) xreduce (F, [a, b, c, d, e]); (%o3) [[[[[("[", simp), a], b], c], d], e]
xreduce
applied to a function not known to be n-ary.
G
is called several times, with two arguments each time.
(%i1) G ([L]) := L; (%o1) G([L]) := L (%i2) xreduce (G, [a, b, c, d, e]); (%o2) [[[[[("[", simp), a], b], c], d], e] (%i3) lreduce (G, [a, b, c, d, e]); (%o3) [[[[a, b], c], d], e]
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