Add basic usage to README

README.md

@@ -3,11 +3,41 @@
 [![Build Status](https://github.com/dpsanders/IntervalBoxes.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/dpsanders/IntervalBoxes.jl/actions/workflows/CI.yml?query=branch%3Amain)
 
 
-Multi-dimensional interval boxes in Julia.
+## Multi-dimensional interval boxes in Julia.
 
-These are Cartesian products of `Interval`s (actually `BareInterval`s) defined in [IntervalArithmetic.jl](https://github.com/JuliaIntervals/IntervalArithmetic.jl).
+An **interval box** is a Cartesian product of `Interval`s (actually `BareInterval`s), as defined in [IntervalArithmetic.jl](https://github.com/JuliaIntervals/IntervalArithmetic.jl).
+An interval box of dimension $n$ thus represents an axis-aligned *set* in Euclidean space $\mathbb{R}^n$.
+
+
+## Basic usage
+
+```jl
+julia> X = IntervalBox(1..3, 4..6)
+[1.0, 3.0] × [4.0, 6.0]
+
+julia> Y = IntervalBox(2..5, 2)
+[2.0, 5.0]²
+
+julia> X ∩ Y
+[2.0, 3.0] × [4.0, 5.0]
+
+julia> X ∪ Y
+[1.0, 5.0] × [2.0, 6.0]
+
+julia> [2, 5] ∈ X ∩ Y
+true
+```
+
+We treat `IntervalBox`es as sets as much as possible, and extend Julia's set operations
+to act on these objects.
+
+Note that the `∪` operator produces the *interval hull* of the union
+(i.e. the smallest interval box that contains the union).
 
-[This code was originally in that repo but was removed. The git history can be found there.]
 
 ## Authors
-Copyright by JuliaIntervals contributors 2017–2024
+Copyright by JuliaIntervals contributors 2017–2024.
+
+This code was originally in `IntervalArithmetic.jl`, but was removed. 
+The git history can be found in that repo.
+