Operations on sets

In this section we show which typical set operations this library supports.

We use the following four sets for illustration.

using LazySets, LazySets.Approximations, Plots
B1 = Ball1(-ones(2), 1.)
B2 = Ball2(ones(2), 1.)
BI = BallInf(zeros(2), 1.)
H = Hyperrectangle(ones(2), ones(2))
sets = [B1, B2, BI, H]

function plot_sets(sets)
    for S in sets
        println(S)
        plot!(S, 1e-2, fillalpha=0.1)
    end
end

function plot_points(points, prefix)
    for i in eachindex(points)
        p = points[i]
        num_occur = length(find(x -> x == p, points[1:i]))
        x = p[1]
        y = p[2]
        if num_occur == 1
            x += 0.15
        elseif num_occur == 2
            y += 0.15
        elseif num_occur == 3
            x -= 0.15
        else
            y -= 0.15
        end
        plot!(Singleton(p))
        plot!(annotations=(x, y, text("$(prefix)$(i)")))
    end
end

plot1 = plot()
plot_sets(sets)
plot1

Unary operations

The following table lists all operations that take one convex set as argument in the columns. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. The table entries have the following meaning.

"x" indicates that the operation is implemented for the respective set type.
"i" indicates that the operation is inherited from a supertype.

type ↓ \ operation →	dim	ρ	σ	an_element	∈	isempty	linear_map	norm	radius	diameter
Interfaces
`LazySet`		x		x						x
`APolytope`		i		i		x	x			i
`ACentrallySymmetric`	x	i		x		x				i
`ACentrallySymmetricPolytope`	i	i		i		x	i			i
`APolygon`	x	i		i		i	i			i
`AHyperrectangle`	i	i	x	i	x	i	i	x	x	i
`AHPolygon`	i	i		x	x	i	i			i
`ASingleton`	i	i	x	i	x	i	x	i	i	i

Basic set types
`Ball1`	i	i	x	i	x	i	i			i
`Ball2`	i	i	x	i	x	i				i
`BallInf`	i	i	i	i	i	i	i	i	x	i
`Ballp`	i	i	x	i	x	i				i
`Ellipsoid`	i	i	x	i	x	i				i
`EmptySet`	x	i	x	x	x	x		x	x	x
`HalfSpace`	x	i	x	x	x	x				i
`HPolygon`/`HPolygonOpt`	i	i	x	i	i	i	i			i
`HPolyhedron`	x	x	x	i	x	x				i
`HPolytope`	x	x	x	i	x	x	i			i
`Hyperplane`	x	i	x	x	x	x				i
`Hyperrectangle`	i	i	i	i	i	i	i	i	i	i
`Interval`	x	i	x	x	x	i	i	i	i	i
`Line`	x	i	x	x	x	x				i
`LineSegment`	x	i	x	x	x	i	i			i
`Singleton`	i	i	i	i	i	i	i	i	i	i
`VPolygon`	i	i	x	x	x	i	x			i
`VPolytope`	x	i	x	i		i	x			i
`ZeroSet`	x	i	x	i	x	i	x	i	i	i
`Zonotope`	i	i	x	i	x	i	x			i

Lazy set operation types
`CartesianProduct`	x	x	x	i	x	x				i
`CartesianProductArray`	x	x	x	i	x	x				i
`ConvexHull`	x	x	x	i		x				i
`ConvexHullArray`	x	x	x	i		x				i
`ExponentialMap`	x	x	x	i	x	x				i
`ExponentialProjectionMap`	x	i	x	i		x				i
`Intersection`	x	x		i	x	x				i
`IntersectionArray`	x	i		i	x					i
`LinearMap`	x	x	x	x	x	x				i
`MinkowskiSum`	x	x	x	i		x				i
`MinkowskiSumArray`	x	x	x	i		x				i
`CacheMinkowskiSum`	x	i	x	i		x				i
`SymmetricIntervalHull`	x	i	x	i	i	i	i	i	i	i

`dim`

This function returns the dimension of the set.

dim(B1), dim(B2), dim(BI), dim(H)

(2, 2, 2, 2)

`ρ`/`σ`

These functions return the support function resp. the support vector of the set.

`an_element`

This function returns some element in the set. Consecutive calls to this function typically return the same element.

an_element(B1), an_element(B2), an_element(BI), an_element(H)

([-1.0, -1.0], [1.0, 1.0], [0.0, 0.0], [1.0, 1.0])

`∈`

This function checks containment of a given vector in the set. The operator can be used in infix notation (v ∈ S) and in inverse operand order (S ∋ v). Alias: in

p1 = [1.5, 1.5]
p2 = [0.1, 0.1]
p3 = [-0.9, -0.8]
points = [p1, p2, p3]

for p in [p1, p2, p3]
    println("$p ∈ (B1, B2, BI, H)? ($(p ∈ B1), $(p ∈ B2), $(p ∈ BI), $(p ∈ H))")
end

[1.5, 1.5] ∈ (B1, B2, BI, H)? (false, true, false, true)
[0.1, 0.1] ∈ (B1, B2, BI, H)? (false, false, true, true)
[-0.9, -0.8] ∈ (B1, B2, BI, H)? (true, false, true, false)

plot1 = plot()
plot_sets(sets)
plot_points(points, "p")
plot1

`isempty`

This function checks if the set is empty.

`linear_map`

This function applies a concrete linear map to the set. The resulting set may be of a different type.

`norm`

This function returns the norm of a set. It is defined as the norm of the enclosing ball (of the given norm) of minimal volume centered in the origin.

# print 1-norm, 2-norm, and infinity norm (if available)
println(("-", "-", norm(B1, Inf)))
println(("-", "-", norm(B2, Inf)))
println((norm(BI, 1), norm(BI, 2), norm(BI, Inf)))
println((norm(H, 1), norm(H, 2), norm(H, Inf)))

("-", "-", 2.0)
("-", "-", 2.0)
(2.0, 1.4142135623730951, 1.0)
(4.0, 2.8284271247461903, 2.0)

`radius`

This function returns the radius of a set. It is defined as the radius of the enclosing ball (of the given norm) of minimal volume with the same center.

radius(B1), radius(B2), radius(BI), radius(H)

(1.0, 1.0, 1.0, 1.0)

`diameter`

This function returns the diameter of a set. It is defined as the diameter of the enclosing ball (of the given norm) of minimal volume with the same center. The implementation is inherited for all set types if the norm is the infinity norm, in which case the result is defined as twice the radius.

diameter(B1), diameter(B2), diameter(BI), diameter(H)

(2.0, 2.0, 2.0, 2.0)

Binary operations

The following table lists all operations that take two convex set as argument in the entries. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. In the columns we also list the operations, but abbreviated. The table entries consist of subsets of the following list of operations.

"⊆" stands for the subset check issubset.
"⊎" stands for the disjointness check isdisjoint.
"∩" stands for the concrete intersection operation intersection.
"C" stands for the conversion operation convert.
"-" indicates that the two types' dimensionality constraints are incompatible.
A suffix "i" indicates that the operation is inherited from a supertype.

type ↓ \ type →	LazyS	APtop	ACSym	ACSPt	APgon	AHrec	AHPgn	ASing	Ball1	Ball2	BInf	Ballp	Ellip	Empty	HalfS	HPgon	HPhed	HPtop	Hplan	Hrect	Itrvl	Line	LineS	Singl	VPgon	VPtop	ZeroS	Zonot	CP	CPA	CH	CHA	EMap	EPM	Itsct	ItscA	LiMap	MS	MSA	CMS	SIH
Interfaces															⊎
`LazySet`	⊎	⊎i	⊎i	⊎i	⊎i	⊆ ⊎i	⊆i ⊎	⊎	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎	⊎	⊆i ⊎	⊆i ⊎i	⊎i	⊎	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i
`APolytope`	⊆ ⊎i	⊆i ⊎i ∩	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆ ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩	⊆i ⊎i ∩	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`ACentrallySymmetric`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`ACentrallySymmetricPolytope`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`APolygon`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	-	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`AHyperrectangle`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆ ⊎ ∩	⊆i ⊎i ∩i	⊆i ⊎ ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`AHPolygon`	⊆i ⊎	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	-	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i C	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`ASingleton`	⊆ ⊎	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆ ⊎ ∩i	⊆i ⊎i ∩i	⊆ ⊎ ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
															⊎
Basic set types															⊎
`Ball1`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Ball2`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆ ⊎i	⊎i	⊆ ⊎	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆ ⊎i	⊎i	⊎i	⊆ ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`BallInf`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Ballp`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆ ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆ ⊎i	⊎i	⊎i	⊆ ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`Ellipsoid`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`EmptySet`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`HalfSpace`	⊎	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎	⊎i	⊎i ∩	⊎i ∩	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`HPolygon`/`HPolygonOpt`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i C	⊆i ⊎i ∩i C	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i C	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i ∩i C	⊆i ⊎i	⊆i ⊎i ∩i Ci	-	⊆i ⊎i	⊆i ⊎i ∩i C	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci
`HPolyhedron`	⊎	⊎i ∩ C	⊎i	⊎i ∩i Ci	⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊎i ∩i Ci	⊎i	⊆i ⊎i ∩i Ci	⊎i	⊎i	⊎i	⊎i ∩	⊎i ∩i Ci	⊎i ∩	⊎i ∩ Ci	⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊎i	⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊎i ∩i Ci	⊎i ∩ Ci	⊆i ⊎i ∩i Ci	⊎i ∩i Ci	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i ∩i Ci
`HPolytope`	⊆i ⊎	⊆i ⊎i ∩ C	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i C	⊆i ⊎i ∩i C	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩	⊆i ⊎i ∩i C	⊆i ⊎i ∩	⊆i ⊎i ∩ Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩ C	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci
`Hyperplane`	⊎	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`Hyperrectangle`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i C	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Interval`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	-	⊆i ⊎i ∩i	-	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	-	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	-	-	⊆i ⊎i ∩i	-	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Line`	⊎	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	-	⊎i ∩	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`LineSegment`	⊆ ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆ ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	-	⊆i ⊎i	⊆i ⊎ ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Singleton`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`VPolygon`	⊆i ⊎i	⊆i ⊎i ∩i C	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i C	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	-	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci
`VPolytope`	⊆i ⊎i	⊆i ⊎i ∩i C	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩	⊆i ⊎i ∩ C	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩ Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci
`ZeroSet`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i
`Zonotope`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i C	⊆i ⊎i ∩i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i Ci	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i Ci	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i Ci
															⊎
Lazy set operation types															⊎
`CartesianProduct`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`CartesianProductArray`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`ConvexHull`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`ConvexHullArray`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`ExponentialMap`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`ExponentialProjectionMap`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`Intersection`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`IntersectionArray`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`LinearMap`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`MinkowskiSum`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`MinkowskiSumArray`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`CacheMinkowskiSum`	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊆i ⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊎i	⊆i ⊎i
`SymmetricIntervalHull`	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i ∩i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i	⊆i ⊎i ∩i

`⊆`

This function checks whether a set is a subset of another set. It can optionally produce a witness if the subset relation does not hold. The operator can be used in infix notation (X ⊆ S). Alias: issubset

println(B1 ⊆ B2)
w1 = ⊆(B1, B2, true)[2]
println(B1 ⊆ BI)
w2 = ⊆(B1, BI, true)[2]
println(B1 ⊆ H)
w3 = ⊆(B1, H, true)[2]
# 'B2 ⊆ B1' is not supported yet
# w11 = ⊆(B2, B1, true)[2]
println(B2 ⊆ BI)
w4 = ⊆(B2, BI, true)[2]
println(B2 ⊆ H)
println(BI ⊆ B1)
w5 = ⊆(BI, B1, true)[2]
println(BI ⊆ B2)
w6 = ⊆(BI, B2, true)[2]
println(BI ⊆ H)
w7 = ⊆(BI, H, true)[2]
println(H ⊆ B1)
w8 = ⊆(H, B1, true)[2]
println(H ⊆ B2)
w9 = ⊆(H, B2, true)[2]
println(H ⊆ BI)
w10 = ⊆(H, BI, true)[2];

false
false
false
false
true
false
false
false
false
false
false

witnesses = [w1, w2, w3, w4, w5, w6, w7, w8, w9, w10]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1

`is_intersection_empty`

This function checks whether the intersection of two sets is empty. It can optionally produce a witness if the intersection is nonempty.

println(is_intersection_empty(BI, H))
w1 = is_intersection_empty(BI, H, true)[2]
# none of the other combinations are supported yet
# is_intersection_empty(B1, B2)
# is_intersection_empty(B1, BI)
# is_intersection_empty(B1, H)
# w2 = is_intersection_empty(B1, H, true)[2]
# is_intersection_empty(B2, BI)
# is_intersection_empty(B2, H)

false

witnesses = [w1]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1