Returns the cartesian product of objects inheriting from class `Set`

.

setproduct(..., simplify = FALSE, nest = FALSE) # S3 method for Set *(x, y)

... | Sets |
---|---|

simplify | logical, if |

nest | logical, if |

x, y |

Either an object of class `ProductSet`

or an unwrapped object inheriting from `Set`

.

The cartesian product of multiple sets, the 'n-ary Cartesian product', is often
implemented in programming languages as being identical to the cartesian product of two sets applied recursively.
However, for sets \(X, Y, Z\),
$$XYZ \ne (XY)Z$$
This is accommodated with the `nest`

argument. If `nest == TRUE`

then \(X*Y*Z == (X × Y) × Z\), i.e. the cartesian
product for two sets is applied recursively. If `nest == FALSE`

then \(X*Y*Z == (X × Y × Z)\) and
the n-ary cartesian product is computed. As it appears the latter (n-ary product) is more common, `nest = FALSE`

is the default. The N-ary cartesian product of \(N\) sets, \(X1,...,XN\), is defined as
$$X1 × ... × XN = \\{(x1,...,xN) : x1 \epsilon X1 \cap ... \cap xN \epsilon XN\\}$$
where \((x1,...,xN)\) is a tuple.

The product of fuzzy and crisp sets first coerces fuzzy sets to crisp sets by finding their support.

Other operators:
`powerset()`

,
`setcomplement()`

,
`setintersect()`

,
`setpower()`

,
`setsymdiff()`

,
`setunion()`

# difference between nesting Set$new(1, 2) * Set$new(2, 3) * Set$new(4, 5)#> {1, 2} × {2, 3} × {4, 5}setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = FALSE) # same as above#> {1, 2} × {2, 3} × {4, 5}setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = TRUE)#> ({1, 2} × {2, 3}) × {4, 5}unnest_set <- setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = FALSE) nest_set <- setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = TRUE) # note the difference when using contains unnest_set$contains(Tuple$new(1, 3, 5))#> [1] TRUE#> [1] TRUE# product of two sets Set$new(-2:4) * Set$new(2:5)#> {-2, -1,...,3, 4} × {2, 3, 4, 5}setproduct(Set$new(1, 4, "a"), Set$new("a", 6))#> {1, 4, a} × {a, 6}setproduct(Set$new(1, 4, "a"), Set$new("a", 6), simplify = TRUE)#> {(1, a), (4, a),...,(4, 6), (a, 6)}# product of two intervals Interval$new(1, 10) * Interval$new(5, 15)#> [1,10] × [5,15]Interval$new(1, 2, type = "()") * Interval$new(2, 3, type = "(]")#> (1,2) × (2,3]Interval$new(1, 5, class = "integer") * Interval$new(2, 7, class = "integer")#> {1,...,5} × {2,...,7}# product of mixed set types Set$new(1:10) * Interval$new(5, 15)#> {1, 2,...,9, 10} × [5,15]Set$new(5, 7) * Tuple$new(6, 8, 7)#> {5, 7} × (6, 8, 7)FuzzySet$new(1, 0.1) * Set$new(2)#> {1(0.1)} × {2}# product of FuzzySet FuzzySet$new(1, 0.1, 2, 0.5) * Set$new(2:5)#> {1(0.1), 2(0.5)} × {2, 3, 4, 5}# product of conditional sets ConditionalSet$new(function(x, y) x >= y) * ConditionalSet$new(function(x, y) x == y)#> {x >= y : x ∈ V, y ∈ V} × {x == y : x ∈ V, y ∈ V}# product of special sets PosReals$new() * NegReals$new()#> ℝ+ × ℝ-