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August 2016


Interval Sets for Julia

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This package represents intervals of an ordered set. For an interval spanning from a to b, all values x that lie between a and b are defined as being members of the interval.

This package is intended to implement a "minimal" foundation for intervals upon which other packages might build. In particular, we encourage type-piracy for the reason that only one interval package can unambiguously define the .. and ± operators (see below).

Currently this package defines one concrete type, Interval. These define the set spanning from a to b, meaning the interval is defined as the set {x} satisfying a ≤ x ≤ b. This is sometimes written [a,b] (mathematics syntax, not Julia syntax) or a..b.

Optionally, Interval{L,R} can represent open and half-open intervals. The type parameters L and R correspond to the left and right endpoint respectively. The notation ClosedInterval is short for Interval{:closed,:closed}, while OpenInterval is short for Interval{:open,:open}. For example, the interval Interval{:open,:closed} corresponds to the set {x} satisfying a < x ≤ b.


You can construct ClosedIntervals in a variety of ways:

julia> using IntervalSets

julia> ClosedInterval{Float64}(1,3)

julia> 0.5..2.5

julia> 1.5±1

Similarly, you can construct OpenIntervals and Interval{:open,:closed}s, and Interval{:closed,:open}:

julia> OpenInterval{Float64}(1,3)
1.0..3.0 (open)

julia> OpenInterval(0.5..2.5)
0.5..2.5 (open)

julia> Interval{:open,:closed}(1,3)
1..3 (open–closed)

The ± operator may be typed as \pm<TAB> (using Julia's LaTeX syntax tab-completion).

Intervals also support the expected set operations:

julia> 1.75  1.5±1  # \in<TAB>; can also use `in`

julia> 0  1.5±1

julia> 1  OpenInterval(0..1)

julia> intersect(1..5, 3..7)   # can also use `a ∩ b`, where the symbol is \cap<TAB>

julia> isempty(intersect(1..5, 10..11))

julia> (0.25..5)  (3..7.4)    # \cup<TAB>; can also use union()

julia> isclosedset(0.5..2.0)

julia> isopenset(OpenInterval(0.5..2.5))

julia> isleftopen(2..3)

When computing the union, the result must also be an interval:

julia> (0.25..5)  (6..7.4)
------ ArgumentError ------------------- Stacktrace (most recent call last)

 [1] — union(::IntervalSets.ClosedInterval{Float64}, ::IntervalSets.ClosedInterval{Float64}) at closed.jl:34

ArgumentError: Cannot construct union of disjoint sets.

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