# Equality of domains

A domain behaves as much as possible like the mathematical set it represents, irrespective of its type. Thus, for example, two domains are considered equal if their membership functions agree.

It is not always possible to realize this intended behaviour in practice. Indeed it may be difficult to discover automatically whether two domains are equal, especially when their types are different. Deciding whether two domains are equal requires supporting implementation, hence the outcome is not always accurate. Still, the principle serves as a design goal.

## The isequaldomain function

Equality of domains is decided by the isequaldomain function in general, and simply by == for subtypes of the Domain supertype.

julia> UnitInterval() == UnitSimplex{Float64}() == 0..1
true

julia> ChebyshevInterval() == UnitBall{Float64}() == -1 .. 1
true

The methodology behind this example is explained in the section on Canonical domains.

Note

If two domains are equal according to ==, then their hashes are also equal. This allows the use of domains as keys in a Dict, if one is so inclined.

DomainSets.isequaldomainFunction
isequaldomain(d1, d2)

Are the two given domains equal?

Domains are considered equal if their membership functions return the same output for the same input.

It is not always possible to verify this automatically. If the result is true, then the domains are guaranteed to be equal. If the result is false, then either the domains are not equal or they are equal but the implementation fails to recognize this.

## Types versus mathematical sets

In some cases the notion of a set is more interesting than the functionality provided by the membership function. A concrete type offers a representation of a notion. Having a type for the real numbers (see ℝ) allows one to specify the domain of a function of a real variable.

A type can also be convenient for other reasons. Although a ball with center and radius can be represented in terms of an affine map and a unit ball, not all affine maps map the unit ball to a ball. Thus, in Julia, such a representation does not allow one to dispatch on the property of being a ball.

Yet, having more types also comes with disadvantages. Dispatching on the type of a domain such as an Interval does not make a function apply to all intervals - because a scalar unit ball has a different type but mathematically is also an interval.

This tension is not resolved in the package. Still, deciding whether two domains are equal is a relevant aspect.