Abstract

We propose a new notation, {/}, to construct and represent the empty set. This is modelled after the corresponding mathematical symbol ‘∅’.

This complements the existing notation for empty tuples, lists, and dictionaries, which use (), [], and {} respectively.

Motivation

Sets are currently the only built-in collection type that have a display syntax, but no notation to express an empty collection. The Python Language Reference notes this, stating:

An empty set cannot be constructed with {}; this literal constructs an empty dictionary.

This can be confusing for beginners, especially those coming to the language from a scientific or mathematical background, where sets may be in more common use than dictionaries or maps.

A syntax notation for the empty set has the important benefit of not requiring a name lookup (unlike set()). {/} will always have a consistent meaning, improving teachability of core concepts to beginners. For example, users must be careful not to use set as a local variable name, as doing so prevents constructing new sets. This can be frustrating as beginners may not know how to recover the set type if they have overriden the name: techniques to do so (e.g. type({1}-{1})) are not immediately obvious.

Finally, this may be helpful for users who do not speak English, as it provides a culture-free notation for a common data structure that is built into Python.

Rationale

Sets were introduced to Python 2.2 via PEP 218, which did not include set notation, but discussed the idea of {-} for the empty set:

The PEP originally proposed {1,2,3} as the set notation and {-} for the empty set. Experience with Python 2.3’s sets.py showed that the notation was not necessary. Also, there was some risk of making dictionaries less instantly recognizable.

Python 3.0 introduced set literals (PEP 3100), but again chose not to introduce notation for the empty set, which was omitted out of pragmatism ([5], [6]).

Since then, the topic has been discussed several times, with various proposals, including:

1. Changing {} to mean an empty set and using {:} for an empty dictionary ([9], [4])
2. A Unicode character (e.g. ∅ or ϕ) ([7])
3. <> ([11], [3])
4. s{} ([10])
5. {*()}, perhaps optimising to compile this to the BUILD_SET opcode ([2])
6. {-} ([8])
7. {,} ([1])
8. {/} ([9])
9. set() (i.e. doing nothing)

The authors propose {/} as the notation, being in their opinion the best of these options. It is simple and concise, with the benefit of resembling the mathematical notation for the empty set, ∅. This makes it an easier mnemonic to explain than several in the language, such as ** for exponentiation or @ for matrix multiplication.

Brief summaries for the other proposals may be found in Rejected Ideas.

Specification

The grammar for set displays will become:

set_display ::=  "{" ("/" | flexible_expression_list | comprehension) "}"

{/} will become the default syntax for the empty set.

>>> type({/})
<class 'set'>
>>> {/} == set()
True

The representation and string forms of the empty set will change to '{/}'.

>>> repr({/})
'{/}'
>>> str({/})
'{/}'

There will be no behavioural changes to set objects.

Backwards Compatibility

Code that relies on the repr() or str() of the empty set will no longer work, because the representation will change.

There will be no other backwards incompatibile changes, all current constructors for the empty set will continue to work, and the behaviour of the set type will remain unchanged.

Security Implications

None.

How to Teach This

Experienced users can be taught that {/} is the new spelling for set(), and that it is equivalent in all other ways.

For new users, sets can be introduced through syntax, noting that the four built-in collection types with syntax all have empty forms: (), [], {/}, and {}.

The empty set uses a forwards slash to differentiate it from an empty dictionary. It uses this syntax because it looks like the mathematical symbol for the empty set (’∅’). It is also similar to the syntax for other sets, which use a comma to separate elements, e.g. {2, 3, 5}. These facts can be used as a helpful mnemonic.

Reference Implementation

Forthcoming.

Rejected Ideas

Open Issues

None.

Footnotes

Copyright

This document is placed in the public domain or under the CC0-1.0-Universal license, whichever is more permissive.