Sets, bags, and mixes

Unordered collections of unique and weighted objects in Perl 6

Table of Contents
1	Introduction
2	Operators with Set semantics
2.1	Set operators that return `Bool`
2.1.1	infix (elem), infix ∈
2.1.2	infix ∉
2.1.3	infix (cont), infix ∋
2.1.4	infix ∌
2.1.5	infix (<=), infix ⊆
2.1.6	infix ⊈
2.1.7	infix (<), infix ⊂
2.1.8	infix ⊄
2.1.9	infix (>=), infix ⊇
2.1.10	infix ⊉
2.1.11	infix (>), infix ⊃
2.1.12	infix ⊅
2.2	Set operators that return a `QuantHash`
2.2.1	infix (\|), infix ∪
2.2.2	infix (&), infix ∩
2.2.3	infix (-), infix ∖
2.2.4	infix (^), infix ⊖
2.3	Set operators that return a `Baggy`
2.3.1	infix (.), infix ⊍
2.3.2	infix (+), infix ⊎
2.4	Terms related to set operators
2.4.1	term ∅

Introduction

The six collection classes are Set, SetHash, Bag, BagHash, Mix and MixHash. They all share similar semantics.

In a nutshell, these classes hold, in general, unordered collections of objects, much like an object hash. The QuantHash role is the role that is implemented by all of these classes: therefore they are also referenced as QuantHashes.

Set and SetHash also implement the Setty role, Bag and BagHash implement the Baggy role, Mix and MixHash implement the Mixy role (which itself implements the Baggy role).

Sets only consider if objects in the collection are present or not, bags can hold several objects of the same kind, and mixes also allow fractional (and negative) weights. The regular versions are immutable, the Hash versions are mutable.

Let's elaborate on that. If you want to collect objects in a container but you do not care about the order of these objects, Perl 6 provides these unordered collection types. Being unordered, these containers can be more efficient than Lists or Arrays for looking up elements or dealing with repeated items.

On the other hand, if you want to get the contained objects (elements) without duplicates and you only care whether an element is in the collection or not, you can use a Set or SetHash.

If you want to get rid of duplicates but still preserve order, take a look at the unique routine for List.

If you want to keep track of the number of times each object appeared, you can use a Bag or BagHash. In these Baggy containers each element has a weight (an unsigned integer) indicating the number of times the same object has been included in the collection.

The types Mix and MixHash are similar to Bag and BagHash, but they also allow fractional and negative weights.

Set, Bag, and Mix are immutable types. Use the mutable variants SetHash, BagHash, and MixHash if you want to add or remove elements after the container has been constructed.

For one thing, as far as they are concerned, identical objects refer to the same element – where identity is determined using the WHICH method (i.e. the same way that the === operator checks identity). For value types like Str, this means having the same value; for reference types like Array, it means referring to the same object instance.

Secondly, they provide a Hash-like interface where the actual elements of the collection (which can be objects of any type) are the 'keys', and the associated weights are the 'values':

type of $a	value of $a{$b} if $b is an element	value of $a{$b} if $b is not an element
Set / SetHash	True	False
Bag / BagHash	a positive integer	0
Mix / MixHash	a non-zero real number	0

Operators with Set semantics

There are several infix operators devoted to performing common operations using QuantHash semantics. Since that is a mouthful, these operators are usually referred to as "set operators".

This does not mean that the parameters of these operators must always be Set, or even a more generic QuantHash. It just means that the logic that is applied to the operators, follows the logic of Set Theory.

These infixes can be written using the Unicode character that represents the function (like ∈ or ∪), or with an equivalent ASCII version (like (elem) or <(|)>.

So explicitly using Set (or Bag or Mix) objects with these infixes is unnecessary. All set operators work with all possible arguments. If necessary, a coercion will take place internally: but in many cases that is not actually needed.

However, if a Bag or Mix is one of the parameters to these set operators, then the semantics will be upgraded to that type (where Mix supersedes Bag if both types happen to be used).

Sets, bags, and mixes

Introduction

Operators with Set semantics

Set operators that return `Bool`

infix (elem), infix ∈

infix ∉

infix (cont), infix ∋

infix ∌

infix (<=), infix ⊆

infix ⊈

infix (<), infix ⊂

infix ⊄

infix (>=), infix ⊇

infix ⊉

infix (>), infix ⊃

infix ⊅

Set operators that return a `QuantHash`

infix (|), infix ∪

infix (&), infix ∩

infix (-), infix ∖

infix (^), infix ⊖

Set operators that return a `Baggy`

infix (.), infix ⊍

infix (+), infix ⊎

Terms related to set operators

term ∅