# Generalized sets Creation and manipulation of generalized sets. ```r gset(support, memberships, charfun, elements, universe, bound, assume_numeric_memberships) as.gset(x) is.gset(x) gset_support(x) gset_core(x, na.rm = FALSE) gset_peak(x, na.rm = FALSE) gset_height(x, na.rm = FALSE) gset_universe(x) gset_bound(x) gset_memberships(x, filter = NULL) gset_transform_memberships(x, FUN, ...) gset_concentrate(x) gset_dilate(x) gset_normalize(x, height = 1) gset_defuzzify(x, method = c("meanofmax", "smallestofmax", "largestofmax", "centroid")) gset_is_empty(x, na.rm = FALSE) gset_is_subset(x, y, na.rm = FALSE) gset_is_proper_subset(x, y, na.rm = FALSE) gset_is_equal(x, y, na.rm = FALSE) gset_contains_element(x, e) gset_is_set(x, na.rm = FALSE) gset_is_multiset(x, na.rm = FALSE) gset_is_fuzzy_set(x, na.rm = FALSE) gset_is_set_or_multiset(x, na.rm = FALSE) gset_is_set_or_fuzzy_set(x, na.rm = FALSE) gset_is_fuzzy_multiset(x) gset_is_crisp(x, na.rm = FALSE) gset_has_missings(x) gset_cardinality(x, type = c("absolute", "relative"), na.rm = FALSE) gset_union(...) gset_sum(...) gset_difference(...) gset_product(...) gset_mean(x, y, type = c("arithmetic", "geometric", "harmonic")) gset_intersection(...) gset_symdiff(...) gset_complement(x, y) gset_power(x) gset_cartesian(...) gset_combn(x, m) e(x, memberships = 1L) is_element(e) ## S3 method for class 'gset' cut(x, level = 1, type = c("alpha", "nu"), strict = FALSE, ...) ## S3 method for class 'gset' mean(x, ..., na.rm = FALSE) ## S3 method for class 'gset' ## median(x, na.rm = FALSE, ...) [R >= 3.4.0] ## median(x, na.rm) [R < 3.4.0] ## S3 method for class 'gset' length(x) ## S3 method for class 'gset' lengths(x, use.names = TRUE) ``` ## Arguments - `x`: For `e()`, `as.gset()` and `is.gset()`: an object. A (g)set object otherwise. `gset_memberships()` also accepts tuple objects. - `y`: A (g)set object. - `e`: An object of class `element`. - `filter`: Optional vector of elements to be filtered. - `m`: Number of elements to choose. - `support`: A set of elements giving the support of the gset (elements with non-zero memberships). Must be a subset of the universe, if specified. - `memberships`: For an (ordinary ) set: 1L (or simply missing). For a fuzzy set: a value between 0 and 1. For a multiset: a positive integer. For a fuzzy multiset: a list of multisets with elements from the unit interval (or a list of vectors interpreted as such). Otherwise, the argument will be transformed using `as.gset`. - `elements`: A set (or list) of `e` objects which are object/memberships-pairs. - `charfun`: A function taking an object and returning the membership. - `bound`: Integer used to compute the absolute complement for (fuzzy) multisets. If `NULL`, defaults to the value of `sets_options("bound")`. If the latter is also `NULL`, the maximum multiplicity will be used in computations. - `assume_numeric_memberships`: When applying `carfun()` to the universe, should numeric memberships (i.e. fuzzy sets or multisets) be assumed (default)? If `FALSE`, fuzzy multisets will be created. - `FUN`: A function, to be applied to a membership vector. - `type`: For `gset_cardinality()`: cardinality type (either `"absolute"` or `"relative"`). For `gset_mean()`: mean type (`"arithmetic"`, `"geometric"`, or `"harmonic"`). For `"cut"`: either `"alpha"` or `"nu"`. - `strict`: Logical indicating whether the cut level must be exceeded strictly (greater than ) or not (greater than or equal ). - `height`: Double from the unit interval for scaling memberships. - `universe`: An optional set of elements. If `NULL`, defaults to the value of `sets_options("universe")`. If the latter is also `NULL`, the support will be used in computations. - `method`: `"centroid"` computes the arithmetic mean of the set elements, using the membership values as weights. `"smallestofmax"` / `"meanofmax"` / `"largestofmax"` returns the minimum/mean/maximum of all set elements with maximal membership degree. - `level`: The minimum membership level. - `na.rm`: logical indicating whether `NA` values should be removed. - `use.names`: logical; should the names of `x` be used in the result? - ``...``: For `gset_foo()`: (g)set objects. For the mean and sort methods: additional parameters internally passed to `mean` and `order`, respectively. For `gset_transform_memberships`: further arguments passed to `FUN`. For `cut`: currently not used. ## Details These functions represent basic infrastructure for handling **generalized** sets of general ( ) objects. A generalized set (or gset) is set of pairs $(e, f)$, where $e$ is some set element and $f$ is the characteristic (or membership) function. For (ordinary ) sets $f$ maps to $\{0, 1\}$, for fuzzy sets into the unit interval, for multisets into the natural numbers, and for fuzzy multisets $f$ maps to the set of multisets over the unit interval. The `gset_is_foo()` predicates are vectorized. In addition to the methods defined, one can use the following operators: `|` for the union, `&` for the intersection, `+` for the sum, `-` for the difference, `%D%` for the symmetric difference, `*` and `^n` for the ($n$-fold) cartesian product, `2^` for the power set, `%e%` for the element-of predicate, `<` and `<=` for the (proper) subset predicate, `>` and `>=` for the (proper) superset predicate, and `==` and `!=` for (in)equality. The `Summary` methods do also work if defined for the set elements. The `mean` and `median` methods try to convert the object to a numeric vector before calling the default methods. `set_combn` returns the gset of all subsets of specified length. `gset_support`, `gset_core`, and `gset_peak` return the set of elements with memberships greater than zero, equal to one, and equal to the maximum membership, respectively. `gset_memberships` returns the membership vector(s) of a given (tuple of) gset(s), optionally restricted to the elements specified by `filter`. `gset_height` returns only the largest membership degree. `gset_cardinality` computes either the absolute or the relative cardinality, i.e. the memberships sum, or the absolute cardinality divided by the number of elements, respectively. The `length` method for gsets gives the (absolute) cardinality. The `lengths` method coerces the set to a list before applying the `length` method on its elements. `gset_transform_memberships` applies function `FOO` to the membership vector of the supplied gset and returns the transformed gset. The transformed memberships are guaranteed to be in the unit interval. `gset_concentrate` and `gset_dilate` are convenience functions, using the square and the square root, respectively. `gset_normalize` divides the memberships by their maximum and scales with `height`. `gset_product` (`gset_mean`) of some gsets compute the gset with the corresponding memberships multiplied (averaged). The `cut` method provides both $\alpha$- and $\nu$-cuts. $\alpha$-cuts filter all elements with memberships greater than (or equal to) `level`---the result, thus, is a crisp (multi)set. $\nu$-cuts select those elements with a **multiplicity** exceeding `level` (only sensible for (fuzzy) multisets). Because set elements are unordered, it is not allowed to use **positional** indexing. However, it is possible to do indexing using element labels or simply the elements themselves (useful, e.g., for subassignment). In addition, it is possible to iterate over **all** elements using `for` and `lapply`/`sapply`. `gset_contains_element` is vectorized in `e`, that is, if `e` is an atomic vector or list, the is-element operation is performed element-wise, and a logical vector returned. Note that, however, objects of class `tuple` are taken as atomic objects to correctly handle sets of tuples. ## See Also `set` for ordinary sets, `gset_outer`, and `tuple` for tuples (vectors ). ## References D. Meyer and K. Hornik (2009), Generalized and customizable sets in R, **Journal of Statistical Software** 31 (2), 1--27. tools:::Rd_expr_doi("10.18637/jss.v031.i02") . ## Examples ```r ## multisets (A <- gset(letters[1:5], memberships = c(3, 2, 1, 1, 1))) (B <- gset(c("a", "c", "e", "f"), memberships = c(2, 2, 1, 2))) rep(B, 2) gset_memberships(tuple(A, B), c("a","c")) gset_union(A, B) gset_intersection(A, B) gset_complement(A, B) gset_is_multiset(A) gset_sum(A, B) gset_difference(A, B) ## fuzzy sets (A <- gset(letters[1:5], memberships = c(1, 0.3, 0.8, 0.6, 0.2))) (B <- gset(c("a", "c", "e", "f"), memberships = c(0.7, 1, 0.4, 0.9))) cut(B, 0.5) A * B A <- gset(3L, memberships = 0.5, universe = 1:5) !A ## fuzzy multisets (A <- gset(c("a", "b", "d"), memberships = list(c(0.3, 1, 0.5), c(0.9, 0.1), gset(c(0.4, 0.7), c(1, 2))))) (B <- gset(c("a", "c", "d", "e"), memberships = list(c(0.6, 0.7), c(1, 0.3), c(0.4, 0.5), 0.9))) gset_union(A, B) gset_intersection(A, B) gset_complement(A, B) ## other operations mean(gset(1:3, c(0.1,0.5,0.9))) median(gset(1:3, c(0.1,0.5,0.9))) ## vectorization list(gset(1, 0.5), gset(2, 2L), gset()) <= gset(1, 2L) ```

gset function