set6 (version 0.1.0)

setproduct: Cartesian Product of Sets

Description

Returns the cartesian product of objects inheriting from class Set.

Usage

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

Arguments

...

Sets

simplify

logical, if TRUE returns the result in its simplest (unwrapped) form, usually a Set otherwise a ProductSet.

nest

logical, if FALSE (default) then will treat any ProductSets passed to ... as unwrapped Sets. See details and examples.

x, y

Set

Value

Either an object of class ProductSet or an unwrapped object inheriting from Set.

Details

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 $X\ast Y\ast Z==(X<U+00D7>Y)<U+00D7>Z$, i.e. the cartesian product for two sets is applied recursively. If nest == FALSE then $X\ast Y\ast Z==(X<U+00D7>Y<U+00D7>Z)$ 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 $$\begin{gathered} X1<U+00D7>...<U+00D7>XN= \\ (x1,...,xN):x1ϵX1\cap ...\cap xNϵXN \end{gathered}$$ where $(x1,...,xN)$ is a tuple.

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

See Also

Other operators: powerset(), setcomplement(), setintersect(), setpower(), setsymdiff(), setunion()

Examples

# NOT RUN {
# difference between nesting
Set$new(1, 2) * Set$new(2, 3) * Set$new(4, 5)
setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = FALSE) # same as above
setproduct(Set$new(1, 2) * Set$new(2, 3), Set$new(4, 5), nest = TRUE)
unnest_set = setproduct(Set$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$new(1,3,5))
nest_set$contains(Tuple$new(Tuple$new(1, 3), 5))

# product of two sets
Set$new(-2:4) * Set$new(2:5)
setproduct(Set$new(1,4,"a"), Set$new("a", 6))
setproduct(Set$new(1,4,"a"), Set$new("a", 6), simplify = TRUE)

# product of two intervals
Interval$new(1, 10) * Interval$new(5, 15)
Interval$new(1, 2, type = "()") * Interval$new(2, 3, type = "(]")
Interval$new(1, 5, class = "integer") *
    Interval$new(2, 7, class = "integer")

# product of mixed set types
Set$new(1:10) * Interval$new(5, 15)
Set$new(5,7) * Tuple$new(6, 8, 7)
FuzzySet$new(1,0.1) * Set$new(2)

# product of FuzzySet
FuzzySet$new(1, 0.1, 2, 0.5) * Set$new(2:5)

# product of conditional sets
ConditionalSet$new(function(x, y) x >= y) *
    ConditionalSet$new(function(x, y) x == y)

# product of special sets
PosReals$new() * NegReals$new()

# }