An interval box is a Cartesian product of Intervals (or BareIntervals), as defined in IntervalArithmetic.jl.
An interval box of dimension $n$ thus represents an axis-aligned set in Euclidean space $\mathbb{R}^n$.

We have defined × to be the Cartesian cross product operator, acting on Intervals and/or
IntervalBoxes.

Set operations

We treat IntervalBoxes as sets as much as possible, and extend Julia's set operations
to act on these objects:

julia> X ⊓ Y
[2.0, 3.0]_trv × [4.0, 5.0]_trv
julia> X ⊔ Y
[1.0, 5.0]_trv × [2.0, 6.0]_trv
julia>using StaticArrays
julia>SVector(2, 5) ∈ X ⊓ Y
true

Note that the ⊔ operator produces the interval hull of the union
(i.e. the smallest interval box that contains the union).

One-dimensional intervals can also be treated as sets in the same way, as follows.
(These set operations are deliberately not defined in IntervalArithmetic.jl.)

julia> x = IntervalBox(1..3)
[1.0, 3.0]¹
julia> y = IntervalBox(2..4)
[2.0, 4.0]¹
julia> x ⊓ y
[2.0, 3.0]¹

Range of multi-dimensional functions

Interval arithmetic allows us to compute an enclosure (in general, an over-estimate)
of the range of a multi-dimensional function. E.g.:

julia>f((x, y)) = x + y;
julia>f(X)
[5.0, 9.0]

Note that in order to use numbers in functions like this, the numbers must be wrapped
in ExactReal to specify that they are exact representations.
Alternatively the complete definition can be annotated with @exact:

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