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39.1 Introduction to Sets | ||
39.2 Definitions 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, the simplification process 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 purposes 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) set (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) set(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) set((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) set((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) set(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: set (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
will signal
an error if any argument is a list. 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], set (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]), set (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 (set (x + y + z, x - y + 4, x + y - 5), lambda ([e], freeof (z, e))); (%o1) {- y + x + 4, y + x - 5} |
The section Definitions for Sets has a complete list of the set functions in Maxima.
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There two ways to to iterate over set members. One way is the use
map
; for example:
(%i1) map (f, set (a, b, c)); (%o1) {f(a), f(b), f(c)} |
The other way is to use for x in s do
(%i1) s: set (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 CRE form. An example is
(%i1) set ([x], [rat (x)]); Maxima encountered a Lisp error: CAR: #:X13129 is not a LIST Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. |
This command 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: CAR: #:A13129 is not a 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 commands
(%i1) orderlessp ([rat(a)], [rat(b)]); Maxima encountered a Lisp error: CAR: #:B13130 is not a 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) set (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 will all set functions as well as with
Maxima functions in general. It's likely, but not certain, that
if all set members are either in CRE form or have been simplified
using ratsimp
, this bug will not manifest.
Maxima's orderless
and ordergreat
mechanisms are
incompatible with the set functions. If you need to use either orderless
or ordergreat
, issue these commands before constructing any sets
and do not use the unorder
command.
Maxima's sign function has a bug that may cause the Kronecker delta function to misbehave; for example:
(%i1) kron_delta (1/sqrt(2), sqrt(2)/2); (%o1) 0 |
The correct value is 1; the bug is related to the sign
bug
(%i1) sign (1/sqrt(2) - sqrt(2)/2); (%o1) pos |
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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Adjoin x to the set a and return a set. Thus adjoin(x, a)
and union(set(x),a)
are equivalent; however, using adjoin
may be somewhat faster than using union
. If a isn't a
set, signal an error.
(%i1) adjoin (c, set (a, b)); (%o1) {a, b, c} (%i2) adjoin (a, set (a, b)); (%o2) {a, b} |
See also disjoin
.
For nonnegative integers n, return the n-th Bell number. If
s
is a set with n
members, belln(n)
is the number
of partitions of s
. For example:
(%i1) makelist (belln (i), i, 0, 6); (%o1) [1, 1, 2, 5, 15, 52, 203] (%i2) is (cardinality (set_partitions (set ())) = belln (0)); (%o2) true (%i3) is (cardinality (set_partitions (set (1, 2, 3, 4, 5, 6))) = belln (6)); (%o3) true |
When n isn't a nonnegative integer, belln(n)
doesn't
simplify.
(%i1) [belln (x), belln (sqrt(3)), belln (-9)]; (%o1) [belln(x), belln(sqrt(3)), belln(- 9)] |
The function belln
threads over equalities, lists, matrices, and
sets.
Return the number of distinct elements of the set a.
(%i1) cardinality (set ()); (%o1) 0 (%i2) cardinality (set (a, a, b, c)); (%o2) 3 (%i3) cardinality (set (a, a, b, c)), simp: false; (%o3) 3 |
In line (%o3), we see that cardinality works correctly even when simplification has been turned off.
Return a set of lists of the form [x_1, ..., x_n]
, where
x_1 in b_1
, ..., x_n in b_n
.
Signal an error when any b_k isn't a set.
(%i1) cartesian_product (set (0, 1)); (%o1) {[0], [1]} (%i2) cartesian_product (set (0, 1), set (0, 1)); (%o2) {[0, 0], [0, 1], [1, 0], [1, 1]} (%i3) cartesian_product (set (x), set (y), set (z)); (%o3) {[x, y, z]} (%i4) cartesian_product (set (x), set (-1, 0, 1)); (%o4) {[x, - 1], [x, 0], [x, 1]} |
Remove x from the set a and return a set.
If x isn't a member of a, return a. Each of the
following do the same thing: disjoin(x, a)
, delete(x, a)
, and
setdifference(a,set(x))
; however, disjoin
is generally
the fastest way to remove a member from a set.
Signal an error if a isn't a set.
Return true
if the sets a and b are disjoint. Signal an
error if either a or b isn't a set.
When n is a nonzero integer, return the set of its divisors. The set of divisors includes the members 1 and n. The divisors of a negative integer are the divisors of its absolute value.
We can verify that 28 is a perfect number.
(%i1) s: divisors(28); (%o1) {1, 2, 4, 7, 14, 28} (%i2) lreduce ("+", args(s)) - 28; (%o2) 28 |
The function divisors works by simplification; you shouldn't need to manually re-evaluate after a substitution. For example:
(%i1) divisors (a); (%o1) divisors(a) (%i2) subst (8, a, %); (%o2) {1, 2, 4, 8} |
The function divisors threads over equalities, lists, matrices, and sets. Here is an example of threading over a list and an equality.
(%i1) divisors ([a, b, c=d]); (%o1) [divisors(a), divisors(b), divisors(c) = divisors(d)] |
Return true
if and only if x is a member of the
set a. Signal an error if a isn't a set.
Return true
if and only if a is the empty set or
the empty list.
(%i1) map (emptyp, [set (), []]); (%o1) [true, true] (%i2) map (emptyp, [a + b, set (set ()), %pi]); (%o2) [false, false, false] |
Return a set of the equivalence classes of s with respect
to the equivalence relation f. The function f should
be a boolean-valued function defined on the cartesian product
of s with s. Further, the function f should
be an equivalence relation; equiv_classes
, however, doesn't
check that it is.
(%i1) equiv_classes (set (a, b, c), lambda ([x, y], is (x=y))); (%o1) {{a}, {b}, {c}} |
Actually, equiv_classes (s, f)
automatically applies the Maxima
function is
after applying the function f; accordingly,
we can restate the previous example more briefly.
(%i1) equiv_classes (set (a, b, c), "="); (%o1) {{a}, {b}, {c}} |
Here is another example.
(%i1) equiv_classes (set (1, 2, 3, 4, 5, 6, 7), lambda ([x, y], remainder (x - y, 3) = 0)); (%o1) {{1, 4, 7}, {2, 5}, {3, 6}} |
The first argument f should be a predicate (a function that evaluates to true, false, or unknown).
Given one set as the second argument,
every (f, a)
returns true
if f(a_i)
returns true
for all a_i in a.
Since sets are unordered, every
is free to evaluate f(a_i)
in any
order. every
may or may not evaluate f for all a_i in a. Because the
order of evaluation isn't specified, the predicate f should
not have side-effects or signal errors for any input.
Given one or more lists as arguments,
every (f, L_1, ..., L_n)
returns true
if 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.
Since lists are ordered, every
evaluates in the order of increasing index.
To use every
on multiple set arguments, they should first be converted
to an ordered sequence so that their relative alignment becomes well-defined.
If the global flag maperror
is true
(the default), all lists
L_1, ..., L_n must have equal lengths - otherwise, every
signals an error.
When maperror
is false, the list arguments are
effectively truncated each to the length of the shortest list.
The Maxima function is
automatically applied after evaluating the
predicate f.
(%i1) every ("=", [a, b], [a, b]); (%o1) true (%i2) every ("#", [a, b], [a, b]); (%o2) false |
When the third argument is max, return the subset of the set or list s for which the real-valued function f takes on its greatest value; when the third argument is min, return the subset for which f takes on its least value.
(%i1) extremal_subset (set (-2, -1, 0, 1, 2), abs, max); (%o1) {- 2, 2} (%i2) extremal_subset (set (sqrt(2), 1.57, %pi/2), sin, min); (%o2) {sqrt(2)} |
Flatten essentially evaluates an expression as if its main operator had been declared n-ary; there is, however, one difference - flatten doesn't recurse into other function arguments. For example:
(%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))) |
Applied to a set, flatten gathers all members of set elements that are sets; for example:
(%i1) flatten (set (a, set (b), set (set (c)))); (%o1) {a, b, c} (%i2) flatten (set (a, set ([a], set (a)))); (%o2) {a, [a]} |
Flatten works correctly when the main operator is a subscripted function
(%i1) flatten (f[5] (f[5] (x))); (%o1) f (x) 5 |
To flatten an expression, the main operator must be defined for zero or more arguments; if this isn't the case, Maxima will halt with an error. Expressions with special representations, for example CRE expressions, can't be flattened; in this case, flatten returns its argument unchanged.
If a is a set, convert a to a list
and apply full_listify
to each list element.
To convert just the top-level operator of a set to a list, see listify.
If a is a list, convert a to a set and apply
fullsetify
to each set member.
(%i1) fullsetify ([a, [a]]); (%o1) {a, {a}} (%i2) fullsetify ([a, f([b])]); (%o2) {a, f([b])} |
In line (%o2), the argument of f
isn't converted to a set
because the main operator of f([b])
isn't a list.
To convert just the top-level operator of a list to a set, see setify.
The identity function evaluates to its argument for all inputs. To
determine if every member of a set is true
, you can use
(%i1) every (identity, [true, true]); (%o1) true |
If the optional second argument len isn't specified, return the set of all partitions of the integer n. When len is specified, return 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. In either case, each partition is a list sorted from greatest to least.
We say a list [a_1, ..., a_m] is a partition of a nonnegative integer n provided (1) each a_i is a nonzero integer and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.
(%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) xprimep(x) := integerp(x) and (x > 1) and primep(x)$ (%i3) subset (s, lambda ([x], every (xprimep, x))); (%o3) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]} |
(Notice that primep(1)
is true in Maxima. This disagrees with
most definitions of prime.)
Return a set containing the elements that are common to the
sets a_1 through a_n. The function intersect
must receive one or more arguments. Signal an error if any of
a_1 through a_n isn't a set. See also intersection.
Return a set containing the elements that are common to the
sets a_1 through a_n. The function intersection
must receive one or more arguments. Signal an error if any of
a_1 through a_n isn't a set. See also intersect.
The Kronecker delta function; kron_delta (x, y)
simplifies to
1 when is(x = y)
is true and it simplifies to zero when
sign (|x - y|)
is pos
. When sign (|x - y|)
is zero
and x - y
isn't a floating point number (neither a double nor
a bfloat), return 0. Otherwise, return a noun form.
The function, kron_delta
is declared to be
symmetric; thus, for example, kron_delta(x, y) - kron_delta(y, x)
simplifies to zero.
Here are a few examples.
(%i1) [kron_delta (a, a), kron_delta (a + 1, a)]; (%o1) [1, 0] (%i2) kron_delta (a, b); (%o2) kron_delta(a, b) |
Assuming that a > b
makes sign (|a - b|)
evaluate to pos
;
thus
(%i1) assume (a > b)$ (%i2) kron_delta (a, b); (%o2) 0 |
If we instead assume that x >= y
, then sign (|x - y|)
evaluates
to pz
; in this case, kron_delta (x, y)
doesn't simplify
(%i1) assume(x >= y)$ (%i2) kron_delta (x, y); (%o2) kron_delta(x, y) |
Finally, since 1/10 - 0.1
evaluates to a floating point
number, we have
(%i1) kron_delta (1/10, 0.1); 1 (%o1) kron_delta(--, 0.1) 10 |
If you want kron_delta (1/10, 0.1)
to evaluate to 1, apply float
.
(%i1) float (kron_delta (1/10, 0.1)); (%o1) 1 |
If a is a set, return a list containing the members of a; when a isn't a set, return a. To convert a set and all of its members to lists, see full_listify.
The function lreduce
(left reduce) extends a 2-arity
function to an n-arity function by composition; an example should
make this clear. When the optional argument init isn't defined, we have
(%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) |
Notice that the function f is first applied to the
leftmost
list elements (thus the name lreduce).
When init is defined, the second argument to the inner most function
evaluation is init; for example:
(%i1) lreduce (f, [1, 2, 3], 4); (%o1) f(f(f(4, 1), 2), 3) |
The function lreduce
makes it easy to find the product or
sum of the elements of a list.
(%i1) lreduce ("+", args (set (a, b))); (%o1) b + a (%i2) lreduce ("*", args (set (1, 2, 3, 4, 5))); (%o2) 120 |
See also See rreduce, See xreduce, and See tree_reduce.
This function is similar to makelist
, but makeset
allows
multiple substitutions. The first argument e is an expression; the
second argument v is a list of variables; and s is a list or
set of values for the variables v. Each member of s must
have the same length as v. We have makeset (e, v, s)
is the set {z | z = substitute(v -> s_i) and s_i in s}
.
(%i1) makeset (i/j, [i, j], [[a, b], [c, d]]); a c (%o1) {-, -} b d (%i2) ind: set (0, 1, 2, 3)$ (%i3) makeset (i^2 + j^2 + k^2, [i, j, k], cartesian_product (ind, ind, ind)); (%o3) {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 22, 27} |
The Moebius function; when n is product of k distinct
primes, moebius(n)
evaluates to (-1)^k; it evaluates to 1 when
n = 1; and it evaluates to 0 for all other positive integers.
The Moebius function threads over equalities, lists, matrices, and
sets.
Return 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 (a_1, ..., a_n)
evaluates to (a_1 + ... + a_n)!/(a_1! ... a_n!)
. Given no
arguments, multinomial()
evaluates to 1. A user may use
minfactorial
to simplify the value returned by multinomial_coeff
;
for example:
(%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 |
When n is a nonnegative integer, return the number of distinct integer partitions of n.
If the optional parameter a has the value list
, return a
list of the number of distinct partitions of 1,2,3, ... , n.
If n isn't a nonnegative integer, return a noun form.
Definition: If n = k_1 + ... + k_m, where k_1 through k_m are distinct positive integers, we call k_1 + ... + k_m a distinct partition of n.
When n is a nonnegative integer, return the number of partitions
of n. If the optional parameter a has the value list
,
return a list of the number of partitions of 1,2,3, ... , n. If n
isn't a nonnegative integer, return a noun form.
(%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) |
For a nonnegative integer n, num_partitions (n)
is equal to
cardinality (integer_partitions (n))
; however, calling num_partitions
is much faster.
Return a list of two sets; the first set is the subset of a for which the predicate f evaluates to false and the second is the subset of a for which f evaluates to true. If a isn't a set, signal an error. See also subset.
(%i1) partition_set (set (2, 7, 1, 8, 2, 8), evenp); (%o1) [{1, 7}, {2, 8}] (%i2) partition_set (set (x, rat(y), rat(y) + z, 1), lambda ([x], ratp(x))); (%o2)/R/ [{1, x}, {y, y + z}] |
Return 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 not deleted before finding the permutations. Thus
(%i1) permutations ([a, a]); (%o1) {[a, a]} (%i2) permutations ([a, a, b]); (%o2) {[a, a, b], [a, b, a], [b, a, a]} |
If a isn't a list or set, signal an error.
When the optional second argument n isn't defined, return the set
of all subsets of the set a.
powerset(a)
has 2^cardinality(a)
members. Given a second argument,
powerset(a,n)
returns the set of all subsets of a that have
cardinality n. Signal an error if a isn't a set;
additionally signal an error if n isn't a positive integer.
The function rreduce
(right reduce) extends a 2-arity
function to an n-arity function by composition; an example should
make this clear. When the optional argument init isn't defined, we have
(%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))) |
Notice that the function f is first applied to the rightmost list elements (thus the name rreduce). When init is defined, the second argument to the inner most function evaluation is init; for example:
(%i1) rreduce (f, [1, 2, 3], 4); (%o1) f(1, f(2, f(3, 4))) |
The function rreduce
makes it easy to find the product or
sum of the elements of a list.
(%i1) rreduce ("+", args (set (a, b))); (%o1) b + a (%i2) rreduce ("*", args (set (1, 2, 3, 4, 5))); (%o2) 120 |
See also See lreduce, See tree_reduce, and See xreduce.
Return a set containing the elements in the set a that are not in the set b. Signal an error if a or b is not a set.
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
.
setequalp
complains when a or b is not a set.
Construct 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
.
Signal an error if a
isn't a list.
Return true if and only if a is a Maxima set. The function
setp
checks that the operator of its argument is set; it doesn't
check that its argument is a simplified set. Thus
(%i1) setp (set (a, a)), simp: false; (%o1) true |
The function setp
could be coded in Maxima as
setp(a) := is (inpart (a, 0) = set)
.
When the optional argument n is defined, return a set of all decompositions of a into n nonempty disjoint subsets. When n isn't defined, return the set of all partitions.
We say a set P is a partition of a set S provided
The empty set is a partition of itself (the conditions 1 and 2 being vacuously true); thus
(%i1) set_partitions (set ()); (%o1) {{}} |
The cardinality of the set of partitions of a set can be found using stirling2
; thus
(%i1) s: set (0, 1, 2, 3, 4, 5)$ (%i2) p: set_partitions (s, 3)$ (%o3) 90 = 90 (%i4) cardinality(p) = stirling2 (6, 3); |
Each member of p
should have 3 members; let's check.
(%i1) s: set (0, 1, 2, 3, 4, 5)$ (%i2) p: set_partitions (s, 3)$ (%o3) {3} (%i4) map (cardinality, p); |
Finally, for each member of p
, the union of its members should
equal s
; again let's check.
(%i1) s: set (0, 1, 2, 3, 4, 5)$ (%i2) p: set_partitions (s, 3)$ (%o3) {{0, 1, 2, 3, 4, 5}} (%i4) map (lambda ([x], apply (union, listify (x))), p); |
The first argument f should be a predicate (a function that evaluates to true, false, or unknown).
Given one set as the second argument,
some (f, a)
returns true
if f(a_i)
returns true
for at least one a_i in a.
Since sets are unordered, some
is free to evaluate f(a_i)
in any
order. some
may or may not evaluate f for all a_i in a. Because the
order of evaluation isn't specified, the predicate f should
not have side-effects or signal errors for any input.
To use some
on multiple set arguments, they should first be converted
to an ordered sequence so that their relative alignment becomes well-defined.
Given one or more lists as arguments,
some (f, L_1, ..., L_n)
returns true
if f(x_1, ..., x_n)
returns true
for at least one x_1, ..., x_n in L_1, ..., L_n, respectively.
some
may or may not evaluate
f for every combination x_1, ..., x_n.
Since lists are ordered, some
evaluates in the order of increasing index.
If the global flag maperror
is true (the default), all lists
L_1, ..., L_n must have equal lengths - otherwise, some
signals an error.
When maperror
is false, the list arguments are
effectively truncated each to the length of the shortest list.
The Maxima function is
is automatically applied after evaluating the
predicate f.
(%i1) some ("<", [a, b, 5], [1, 2, 8]); (%o1) true (%i2) some ("=", [2, 3], [2, 7]); (%o2) true |
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.
We use a recursion relation to define stirling1 (n, m)
for
m less than 0; we do not extend it for n less than 0 or for non-integer
arguments.
The function stirling1
works by simplification; it knows the
basic special values (see Donald Knuth, The Art of Computer Programming,
third edition, Volume 1, Section 1.2.6, Equations 48, 49, and 50).
For Maxima to apply these rules, the arguments must be declared to
be integer and the first argument must nonnegative. For example:
(%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 knows a few other special values; for example:
(%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! |
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.
We use a recursion relation to define stirling2 (n, m)
for
m less than 0; we do not extend it for n less than 0 or for non-integer
arguments.
The function stirling2
works by simplification; it knows the
basic special values (see Donald Knuth, The Art of Computer Programming,
third edition, Volume 1, Section 1.2.6, Equations 48, 49, and 50).
For Maxima to apply these rules, the arguments must be declared to
be integer and the first argument must nonnegative. For example:
(%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 knows a few other special values.
(%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 |
Return the subset of the set a that satisfies the predicate f. For example:
(%i1) subset (set (1, 2, x, x + y, z, x + y + z), atom); (%o1) {1, 2, x, z} (%i2) subset (set (1, 2, 7, 8, 9, 14), evenp); (%o2) {2, 8, 14} |
The second argument to subset
must be a predicate
(a boolean-valued function of one argument) if the first argument to
subset
isn't a set, signal an error. See also
partition_set.
Return true if and only if the set a is a subset of b. Signal an error if a or b is not a set.
Return the set of members that occur in exactly one
set a_k. Signal an error if any argument a_k isn't a
set. Given two arguments, symmdifference (a, b)
is
the same as union (setdifference (a, b), setdifference (b, a))
.
The function tree_reduce
extends a associative binary operator f : S x S -> S
from two arguments to any number of arguments using a minimum
depth tree. An example should make this clear.
(%i1) tree_reduce (f, [a, b, c, d]); (%o1) f(f(a, b), f(c, d)) |
Given an odd number of arguments, tree_reduce
favors the left
side of the tree; for example:
(%i1) tree_reduce (f, [a, b, c, d, e]); (%o1) f(f(f(a, b), f(c, d)), e) |
For addition of floating point numbers, using tree_reduce
may
give a sum that has a smaller rounding error than using either
rreduce
or lreduce
.
Return the union of the sets a_1 through a_n.
When union
receives no arguments, it returns the
empty set. Signal an error when one or more arguments to
union
is not a set.
This function is similar to both lreduce
and rreduce
except
that xreduce
is free to use either left or right associativity;
in particular when f is an associative function and Maxima
has a built-in evaluator for it, xreduce
may use the n-ary
function; these n-ary functions include
addition +
, multiplication *
, and
, or
, max
,
min
, and append
. For these operators, we generally expect
using xreduce
to be faster than using either rreduce
or
lreduce
. When f isn't n-ary, xreduce
uses
left-associativity.
Floating point addition is not associative; nevertheless, xreduce
uses Maxima's n-ary addition when the set or list s contains
floating point numbers.
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