## Maxima – 196 – Insiemi – 5

Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.593.

`intersect (a_1, ..., a_n)`
`intersect` is the same as intersection.

`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.

``````(%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)                                 {}``````

`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.

``````(%i9) kron_delta(a,a);
(%o9)                                  1
(%i10) kron_delta(a,b,a,b);
(%o10)                         kron_delta(a, b)
(%i11) kron_delta(a,a,b,a+1);
(%o11)                                 0
(%i12) assume(equal(x,y));
(%o12)                           [equal(x, y)]
(%i13) kron_delta(x,y);
(%o13)                                 1``````

`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.

``````(%i14) listify ({a, b, c, d});
(%o14)                           [a, b, c, d]
(%i15) listify (F ({a, b, c, d}));
(%o15)                          F({a, b, c, d})``````

`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`.

``````(%i16) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
1  2  3  4
(%o16)                           {-, -, -, -}
a  b  c  d
(%i17) S : {x, y, z}\$

(%i18) S3 : cartesian_product (S, S, S);
(%o18) {[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]}
(%i19) makeset (i + j + k, [i, j, k], S3);
(%o19) {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}
(%i20) makeset (sin(x), [x], {[1], [2], [3]});
(%o20)                     {sin(1), sin(2), sin(3)}``````

`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.

``````(%i22) moebius (1);
(%o22)                                 1
(%i23) moebius (2 * 3 * 5);
(%o23)                                - 1
(%i24) moebius (11 * 17 * 29 * 31);
(%o24)                                 1
(%i25) moebius (2^32);
(%o25)                                 0
(%i26) moebius (n);
(%o26)                            moebius(n)
(%i27) moebius (n = 12);
(%o27)                          moebius(n) = 0
(%i28) moebius ([11, 11 * 13, 11 * 13 * 15]);
(%o28)                            [- 1, 1, 1]
(%i29) moebius (matrix ([11, 12], [13, 14]));
[ - 1  0 ]
(%o29)                            [        ]
[ - 1  1 ]
(%i30) moebius ({21, 22, 23, 24});
(%o30)                            {- 1, 0, 1}``````

`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`.

``````(%i31) multinomial_coeff (1, 2, x);
(x + 3)!
(%o31)                             --------
2 x!
(%i32) minfactorial (%);
(x + 1) (x + 2) (x + 3)
(%o32)                      -----------------------
2
(%i33) multinomial_coeff (-6, 2);

factorial: factorial of negative integer -6 not defined.
-- an error. To debug this try: debugmode(true);``````

OOPS! diverso dal Reference 😐

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