With the introduction to set theory fundamentals in the previous article in this series, we are all set to explore the advanced realms of set theory through Maxima.

**More set operations**

We have already worked out the basic set creation techniques and some basic set operations provided by Maxima. Here are some next-level set operations it provides:

*makeset(expr, vars, varslist)*-Sophisticated set-creation using expressions*adjoin(x, S)*- Returns a set with all elements of S and the element x*disjoin(x, S)*- Returns a set with all elements S but without element x*powerset(S)*- Returns the set of all subsets of S*subset(S, p)*- Returns the subset of S, elements of which satisfy the predicate p*symmdifference(S1, S2)*- Returns the symmetric difference between the sets S1 and S2, i.e., the elements in S1 or S2 but not in both

And here is a demonstration of each one of these operations:

$ maxima -q (%i1) makeset(a+b, [a, b], [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6]]); (%o1) {3, 5, 7, 9, 11} (%i2) makeset(a-b, [a, b], [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6]]); (%o2) {- 1} (%i3) makeset(a*b, [a, b], [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6]]); (%o3) {2, 6, 12, 20, 30} (%i4) makeset(a + 2*a*b + b, [a, b], [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6]]); (%o4) {7, 17, 31, 49, 71} (%i5) quit(); $ maxima -q (%i1) S: {-4, 6, 7, 32, 0}; (%o1) {- 4, 0, 6, 7, 32} (%i2) adjoin(3, S); (%o2) {- 4, 0, 3, 6, 7, 32} (%i3) adjoin(7, S); (%o3) {- 4, 0, 6, 7, 32} (%i4) S: adjoin(3, S); /* Updating S */ (%o4) {- 4, 0, 3, 6, 7, 32} (%i5) adjoin(7, S); (%o5) {- 4, 0, 3, 6, 7, 32} (%i6) disjoin(7, S); (%o6) {- 4, 0, 3, 6, 32} (%i7) disjoin(5, S); (%o7) {- 4, 0, 3, 6, 7, 32} (%i8) quit(); $ maxima -q (%i1) S: {-4, 0, 3, 6, 7, 32}; (%o1) {- 4, 0, 3, 6, 7, 32} (%i2) S1: subset(S, evenp); (%o2) {- 4, 0, 6, 32} (%i3) powerset(S1); (%o3) {{}, {- 4}, {- 4, 0}, {- 4, 0, 6}, {- 4, 0, 6, 32}, {- 4, 0, 32}, {- 4, 6}, {- 4, 6, 32}, {- 4, 32}, {0}, {0, 6}, {0, 6, 32}, {0, 32}, {6}, {6, 32}, {32}} (%i4) S2: {-35, -26, 0, 7, 32, 100}; (%o4) {- 35, - 26, 0, 7, 32, 100} (%i5) symmdifference(S1, S2); (%o5) {- 35, - 26, - 4, 6, 7, 100} (%i6) symmdifference(S, S2); (%o6) {- 35, - 26, - 4, 3, 6, 100} (%i7) quit();

**Advanced set operations**

With Maxima, much more than this can be done with sets, using just the advanced functionalities provided by it. So now, lets take a journey through them.

**Cartesian product:** Given n sets, the function* cartesian_product()* returns a set of lists formed by the Cartesian product of the n sets. The following demonstration explains what this means:

$ maxima -q (%i1) cartesian_product({0, 1, 2}, {a, b, c}); (%o1) {[0, a], [0, b], [0, c], [1, a], [1, b], [1, c], [2, a], [2, b], [2, c]} (%i2) cartesian_product({0, 1}, {a, b}, {X, Y}); (%o2) {[0, a, X], [0, a, Y], [0, b, X], [0, b, Y], [1, a, X], [1, a, Y], [1, b, X], [1, b, Y]} (%i3) cartesian_product({0, 1}, {a, b, c}); (%o3) {[0, a], [0, b], [0, c], [1, a], [1, b], [1, c]} (%i4) quit();

**Set partitions:** Given a set S, it can be partitioned into various subsets, based on various mathematical principles. Maxima provides a host of functions for such partitioning the basic one being *set_partitions()*. It returns a set of all possible partitions of the given set. With a number as the second argument, it gives only the partitions with that exact number of sets in each partition. Shown below are some examples to make sense of this concept:

$ maxima -q (%i1) S: {a, b, c}; (%o1) {a, b, c} (%i2) set_partitions(S); (%o2) {{{a}, {b}, {c}}, {{a}, {b, c}}, {{a, b}, {c}}, {{a, b, c}}, {{a, c}, {b}}} (%i3) set_partitions(S, 1); (%o3) {{{a, b, c}}} (%i4) set_partitions(S, 2); (%o4) {{{a}, {b, c}}, {{a, b}, {c}}, {{a, c}, {b}}} (%i5) set_partitions(S, 3); (%o5) {{{a}, {b}, {c}}} (%i6) set_partitions(S, 4); (%o6) {} (%i7) belln(3); (%o7) 5 (%i8) cardinality(set_partitions(S)); /* Number of elements */ (%o8) 5 (%i9) belln(4); (%o9) 15 (%i10) belln(5); (%o10) 52 (%i11) belln(6); (%o11) 203 (%i12) quit();

In the above examples, *belln()* or the nth Bell number is the number of partitions of a set with n members.

*integer_partitions(n)* is a specific function, which partitions a given positive integer n into a set of positive integers, the sum of which adds up to the original integer. *num_partitions(n)* returns the number of such partitions returned by* integer_partitions(n)*. Examples follow:

$ maxima -q (%i1) integer_partitions(1); (%o1) {[1]} (%i2) num_partitions(1); (%o2) 1 (%i3) integer_partitions(2); (%o3) {[1, 1], [2]} (%i4) num_partitions(2); (%o4) 2 (%i5) integer_partitions(3); (%o5) {[1, 1, 1], [2, 1], [3]} (%i6) num_partitions(3); (%o6) 3 (%i7) integer_partitions(4); (%o7) {[1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]} (%i8) num_partitions(4); (%o8) 5 (%i9) integer_partitions(0); (%o9) {[]} (%i10) num_partitions(0); (%o10) 1 (%i11) integer_partitions(5, 1); (%o11) {[5]} (%i12) integer_partitions(5, 2); (%o12) {[3, 2], [4, 1], [5, 0]} (%i13) integer_partitions(5, 3); (%o13) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]} (%i14) integer_partitions(5, 4); (%o14) {[2, 1, 1, 1], [2, 2, 1, 0], [3, 1, 1, 0], [3, 2, 0, 0], [4, 1, 0, 0], [5, 0, 0, 0]} (%i15) integer_partitions(5, 5); (%o15) {[1, 1, 1, 1, 1], [2, 1, 1, 1, 0], [2, 2, 1, 0, 0], [3, 1, 1, 0, 0], [3, 2, 0, 0, 0], [4, 1, 0, 0, 0], [5, 0, 0, 0, 0]} (%i16) num_partitions(5); (%o16) 7 (%i17) num_distinct_partitions(5); (%o17) 3 (%i18) quit();

Note that like *set_partitions(), integer_partitions()* also takes an optional second argument, limiting the partitions to partitions of cardinality equal to that number. However, note that all smaller-sized partitions are made equal to the corresponding size by adding the required number of zeroes.

Also,* num_distinct_partitions(n)* returns the number of distinct integer partitions of n, i.e., the integer partitions of n with only distinct integers.

Another powerful partitioning function is *equiv_classes(S, r)*, which returns a partition of S, elements of which satisfy the binary relation r. Here are a few examples:

$ maxima -q (%i1) equiv_classes({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, lambda([x, y], remainder(x - y, 2) = 0)); (%o1) {{0, 2, 4, 6, 8, 10}, {1, 3, 5, 7, 9}} (%i2) equiv_classes({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, lambda([x, y], remainder(x - y, 3) = 0)); (%o2) {{0, 3, 6, 9}, {1, 4, 7, 10}, {2, 5, 8}} (%i3) equiv_classes({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, lambda([x, y], remainder(x - y, 5) = 0)); (%o3) {{0, 5, 10}, {1, 6}, {2, 7}, {3, 8}, {4, 9}} (%i4) equiv_classes({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, lambda([x, y], remainder(x - y, 6) = 0)); (%o4) {{0, 6}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {5}} (%i5) equiv_classes({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, lambda([x, y], remainder(x, y) = 0)); (%o5) {{1, 2, 4, 8}, {3, 6}, {5, 10}, {7}, {9}} (%i6) quit();

Notice the relation being defined using *lamda* for the property of divisibility by 2, 3, 5, 6, and among the set elements themselves, respectively.

A closely related function *partition_set*(S, p) partitions S into two sets, one with elements satisfying the predicate p, and the other not satisfying the predicate p. A small demonstration follows:

$ maxima -q (%i1) partition_set({-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 26, 37, 100}, primep); (%o1) [{- 1, 0, 1, 4, 6, 8, 9, 10, 26, 100}, {2, 3, 5, 7, 11, 19, 37}] (%i2) quit();

**Miscellaneous:** And, finally, lets look at some general but mathematically interesting operations:

*divisors(n) –*returns the set of positive divisors of n*permutations(S)*- returns the set of all permutations of the elements of S*random_permutation(S)*- returns one of the elements of permutations(S), randomly*extremal_subset(S, f, max | min)*- returns the subset of S, for which the value of the function f is maximum or minimum

A demonstration of all the functions mentioned above, follows:

$ maxima -q (%i1) divisors(9); (%o1) {1, 3, 9} (%i2) divisors(28); (%o2) {1, 2, 4, 7, 14, 28} (%i3) permutations({a, b, c}); (%o3) {[a, b, c], [a, c, b], [b, a, c], [b, c, a], [c, a, b], [c, b, a]} (%i4) random_permutation({a, b, c}); (%o4) [c, b, a] (%i5) random_permutation({a, b, c}); (%o5) [c, a, b] (%i6) random_permutation({a, b, c}); (%o6) [b, c, a] (%i7) extremal_subset({-5, -3, -1, 0, 1, 2, 3, 4, 5}, lambda([x], x*x), max); (%o7) {- 5, 5} (%i8) extremal_subset({-5, -3, -1, 0, 1, 2, 3, 4, 5}, lambda([x], x*x), min); (%o8) {0} (%i9) quit();

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