Maxima 5.41.0 Manual: 35. Sets

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# 35. Sets

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## 35.1 Introduction to Sets

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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### 35.1.1 Usage

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.

·

@ref{Category: Sets}

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### 35.1.2 Set Member Iteration

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`.

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### 35.1.3 Authors

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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## 35.2 Functions and Variables for Sets

Function: adjoin (x, a)

  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}

  ·

  @ref{Category: Sets}

Function: belln (n)

  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)]

  ·

  @ref{Category: Sets}

Function: cardinality (a)

  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

  ·

  @ref{Category: Sets}

Function: cartesian_product (b_1, ... , b_n)

  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]}

  ·

  @ref{Category: Sets}

Function: disjoin (x, a)

  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}

  ·

  @ref{Category: Sets}

Function: disjointp (a, b)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: divisors (n)

  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)}

  ·

  @ref{Category: Integers}

Function: elementp (x, a)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: emptyp (a)

  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]

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: equiv_classes (s, F)

  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}}

  ·

  @ref{Category: Sets}

Function: every
`every` (f, s)
`every` (f, L_1, ..., L_n)

  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

  ·

  @ref{Category: Sets}

Function: extremal_subset
`extremal_subset` (s, f, max)
`extremal_subset` (s, f, min)

  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)}

  ·

  @ref{Category: Sets}

Function: flatten (expr)

  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);

  ·

  @ref{Category: Sets} · @ref{Category: Lists}

Function: full_listify (a)

  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])]))

  ·

  @ref{Category: Sets}

Function: fullsetify (a)

  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])}

  ·

  @ref{Category: Lists}

Function: identity (x)

  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

Function: integer_partitions
`integer_partitions` (n)
`integer_partitions` (n, len)

  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]}

  ·

  @ref{Category: Integers}

Function: intersect (a_1, ..., a_n)

  `intersect` is the same as `intersection`, which see.

  ·

  @ref{Category: Sets}

Function: intersection (a_1, ..., a_n)

  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)                          {}

  ·

  @ref{Category: Sets}

Function: kron_delta (x1, x2, …, xp)

  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

Function: listify (a)

  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})

  ·

  @ref{Category: Sets}

Function: makeset (expr, x, s)

  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)}

  ·

  @ref{Category: Sets}

Function: moebius (n)

  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}

  ·

  @ref{Category: Integers}

Function: multinomial_coeff
`multinomial_coeff` (a_1, ..., a_n)
`multinomial_coeff` ()

  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

  ·

  @ref{Category: Integers}

Function: num_distinct_partitions
`num_distinct_partitions` (n)
`num_distinct_partitions` (n, list)

  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)

  ·

  @ref{Category: Integers}

Function: num_partitions
`num_partitions` (n)
`num_partitions` (n, list)

  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)

  ·

  @ref{Category: Integers}

Function: partition_set (a, f)

  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}]

  ·

  @ref{Category: Sets}

Function: permutations (a)

  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]}

  ·

  @ref{Category: Sets} · @ref{Category: Lists}

Function: powerset
`powerset` (a)
`powerset` (a, n)

  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)                         {{}}

  ·

  @ref{Category: Sets}

Function: random_permutation (a)

  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]

  ·

  @ref{Category: Sets} · @ref{Category: Lists}

Function: setdifference (a, b)

  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)                          {}

  ·

  @ref{Category: Sets}

Function: setequalp (a, b)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: setify (a)

  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}

  ·

  @ref{Category: Lists}

Function: setp (a)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: set_partitions
`set_partitions` (a)
`set_partitions` (a, n)

  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

  1. each member of P is a nonempty set,
  2. distinct members of P are disjoint,
  3. the union of the members of P equals S.

  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}}

  ·

  @ref{Category: Sets}

Function: some
`some` (f, a)
`some` (f, L_1, ..., L_n)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Lists}

Function: stirling1 (n, m)

  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:

  1. stirling1(1,k) = kron_delta(1,k), k >= 0,(see http://dlmf.nist.gov/26.8.E2)
  2. stirling1(n,n) = 1, n >= 0 (see http://dlmf.nist.gov/26.8.E1)
  3. stirling1(n,n-1) = -binomial(n,2), n >= 1, (see http://dlmf.nist.gov/26.8.E16)
  4. stirling1(n,0) = kron_delta(n,0), n >=0 (see http://dlmf.nist.gov/26.8.E14 and http://dlmf.nist.gov/26.8.E1)
  5. stirling1(n,1) =(-1)^(n-1) (n-1)!, n >= 1 (see http://dlmf.nist.gov/26.8.E14)
  6. stirling1(n,k) = 0, n >= 0 and k > n.

  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

  ·

  @ref{Category: Integers}

Function: stirling2 (n, m)

  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.

  1. stirling2(n,0) = 1, n >= 1 (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,0) = 1)
  2. stirling2(n,n) = 1, n >= 0, (see http://dlmf.nist.gov/26.8.E4)
  3. stirling2(n,1) = 1, n >= 1, (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,1) = 0)
  4. stirling2(n,2) = 2^(n-1) -1, n >= 1, (see http://dlmf.nist.gov/26.8.E17)
  5. stirling2(n,n-1) = binomial(n,2), n>= 1 (see http://dlmf.nist.gov/26.8.E16)
  6. stirling2(n,k) = 0, n >= 0 and k > n.

  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)

  ·

  @ref{Category: Integers}

Function: subset (a, f)

  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}

  ·

  @ref{Category: Sets}

Function: subsetp (a, b)

  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

  ·

  @ref{Category: Sets} · @ref{Category: Predicate functions}

Function: symmdifference (a_1, …, a_n)

  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}

  ·

  @ref{Category: Sets}

Function: union (a_1, ..., a_n)

  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}

  ·

  @ref{Category: Sets}

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