Sets

Sets are unordered, uniqued collections of things. One way to represent sets is by placing (representations of) their elements between curly braces.

For instance, we can represent the set of vowel phonemes in English in the following way.

\[V_1 \equiv \{\text{e, i, o, u, æ, ɑ, ɔ, ə, ɛ, ɪ, ʊ}\}\]

We express that something is an element of a set using the notation \(\cdot \in \cdot\) and that it is not an element using the notation \(\cdot \not\in \cdot\). So for instance, \(\text{e}\) is an element of \(V_1\), while \(\text{t}\) is not.

\[\text{e} \in V_1 \quad \text{t} \not\in V_1\]

To work with sets in Python, we can use the standard library’s set type. These objects function as we would expect in terms of elementhood.

vowels_1: set[str] = {"e", "i", "o", "u", "æ", "ɑ", "ɔ", "ə", "ɛ", "ɪ", "ʊ"}

if "e" in vowels_1:
    print("e ∈ V_1")
else:
    print("e ∉ V_1")
    
if "t" in vowels_1:
    print("t ∈ V_3")
else:
    print("t ∉ V_3")

e ∈ V_1
t ∉ V_3

Sets can be represented in many ways. For instance, we could also represent the set \(V_1\) in this way:

\[V_2 \equiv \{\text{o, u, æ, ɑ, ɔ, ə, ɛ, ɪ, ʊ, e, i}\}\]

Because sets are unordered, both \(V_1\) and \(V_2\) are representations of the exact same set (\(V_1 = V_2\)). And because sets are uniqued, \(V_3\) is also a representation of the same set as \(V_1\) and \(V_2\)–i.e. \(V_1=V_2=V_3\)–even though there are multiple copies of some vowels in this representation.

\[V_3 \equiv \{\text{o, o, o, u, u, æ, ɑ, ɔ, ə, ə, ə, ɛ, ɪ, ʊ, e, i}\}\]

Python sets work as we would expect in terms of equality.

vowels_2: set[str] = {"o", "u", "æ", "ɑ", "ɔ", "ə", "ɛ", "ɪ", "ʊ", "e", "i"}
vowels_3: set[str] = {"o", "o", "o", "u", "u", "æ", "ɑ", "ɔ", "ə", "ə", "ə", "ɛ", "ɪ", "ʊ", "e", "i"}

if vowels_1 == vowels_2:
    print("V_1 = V_2")
else:
    print("V_1 ≠ V_2")
    
if vowels_1 == vowels_3:
    print("V_1 = V_3")
else:
    print("V_1 ≠ V_3")

V_1 = V_2
V_1 = V_3

I’ll just call this set \(V \equiv V_1 = V_2 = V_3\) moving forward.

vowels: set[str] = vowels_1

Python has another way of representing sets that we will have reason to use: frozenset. These work very similarly to sets in a lot of ways.

vowels_frozen: frozenset[str] = frozenset(vowels)

if vowels == vowels_frozen:
    print("V = V_frozen")
else:
    print("V ≠ V_frozen")

V = V_frozen

One big difference between the two is that sets are mutable, while frozensets are immutable. Basically, we can alter sets, but we can’t alter frozensets. For instance, we can add elements to a set but not a frozenset.

try:
    vowels.add("t")
    print("Successfully added 't' to vowels.")
except AttributeError:
    print("Failed to add 't' to vowels.")

try:
    vowels_frozen.add("t")
    print("Successfully added 't' to vowels_frozen.")
except AttributeError:
    print("Failed to add 't' to vowels_frozen.")

Successfully added 't' to vowels.
Failed to add 't' to vowels_frozen.

Similarly, we can remove elements from sets but not frozensets.

try:
    vowels.remove("t")
    print("Successfully removed 't' to vowels.")
except AttributeError:
    print("Failed to remove 't' to vowels.")

try:
    vowels_frozen.remove("ə")
    print("Successfully removed 'ə' to vowels_frozen.")
except AttributeError:
    print("Failed to remove 'ə' to vowels_frozen.")

Successfully removed 't' to vowels.
Failed to remove 'ə' to vowels_frozen.

This behavior makes frozensets seem pretty useless, since it would seem they can do fewer things with them. But frozensets turn out to have a really useful property: they can be elements of other sets or frozensets.

try:
    vowels_singleton: set[set[str]] = {vowels}
    print("Successfully constructed the set {V}.")
except TypeError:
    print("Failed to construct the set {V}.")

try:
    vowels_frozen_singleton: set[frozenset[str]] = {vowels_frozen}
    print("Successfully constructed the set {V_frozen}.")
except TypeError:
    print("Failed to construct the set {V_frozen}.")

Failed to construct the set {V}.
Successfully constructed the set {V_frozen}.

The reason frozensets can be elements of sets or frozensets is that they are hashable, while sets are not.¹ We are going to use this property extensively throughout this course.

Footnotes

1. There is a correlation between hashability and immutability, but they are not the same thing.