Mathematical Set Operations

15 min · Beginner

Python sets natively support mathematical set algebra. Operators like |, &, -, and ^ solve complex comparison and filtering tasks in a single line.

Union (|): all elements combined

Union is the set of every value that is in the first set, the second set, or both. Each value appears once. You write it as a | b.

The union of two sets contains all elements that appear in either set without duplicates.

Python

frontend = {"HTML", "CSS", "JavaScript"}backend = {"Python", "SQL", "JavaScript"}fullstack = frontend | backendprint(sorted(fullstack))

Example 1 · Merging two mailing lists

Python

team_a = {"Asha", "Ravi", "Meena"}team_b = {"Ravi", "Kiran"}everyone = team_a | team_bprint(sorted(everyone))print(len(everyone))

Output

['Asha', 'Kiran', 'Meena', 'Ravi']
4

What this code does

| joins the two sets into one.
Ravi is in both lists, but a set keeps him once.
So the combined list has 4 people, not 5.

The duplicate 'JavaScript' is included only once.

Intersection (&): shared elements

Intersection is the set of values that are in both sets at the same time. You write it as a & b, and it is empty when the sets share nothing.

The intersection contains only elements that exist in both sets.

Python

shared = frontend & backendprint(shared)

Example 2 · Customers who bought both products

Python

bought_phone = {"C1", "C2", "C3", "C4"}bought_case = {"C3", "C4", "C5"}both = bought_phone & bought_caseprint(sorted(both))print(len(both))

Output

['C3', 'C4']
2

What this code does

& keeps only the ids found in both sets.
Customers C3 and C4 bought the phone and the case.
This replaces a nested loop with one short line.

The console displays {'JavaScript'}.

Difference (-) and Symmetric Difference (^)

Set difference a - b is the set of values in a that are not in b. Symmetric difference a ^ b is the values in exactly one of the two sets.

A - B yields elements in A but not in B. A ^ B yields elements in either A or B, but not both.

Python

only_frontend = frontend - backendunique_to_one = frontend ^ backendprint(sorted(only_frontend))print(sorted(unique_to_one))

Example 3 · Students who have not submitted

Python

registered_students = {"Rohan", "Aditi", "Meera", "Dev"}submitted_assignment = {"Aditi", "Dev"}
pending_students = registered_students - submitted_assignmentprint("Pending submission:", sorted(pending_students))

Output

Pending submission: ['Meera', 'Rohan']

What this code does

Subtracts submitted students from registered students using -.
Identifies students who have not yet submitted: ['Meera', 'Rohan'].

Symmetric difference removes the common element 'JavaScript'.

Subset, superset and disjoint

Subset is a set whose values are all found inside another set. a <= b is True when a is a subset of b. Two sets are disjoint when they have no value in common.

Use <= to test a subset, >= for a superset, and isdisjoint() to test that two sets share nothing.

Python

required = {"read", "write"}granted = {"read", "write", "delete"}print(required <= granted)print(granted <= required)print(required.isdisjoint({"admin"}))

Example 4 · Checking permissions

Python

required = {"read", "write", "delete"}granted = {"read", "write"}print(required <= granted)print(sorted(required - granted))print(granted.isdisjoint({"admin"}))

Output

False
['delete']
True

What this code does

required <= granted is False because delete is not granted.
required - granted lists exactly what is missing.
isdisjoint is True because granted has no admin.

Both required permissions are inside granted, so the first test is True. The reverse is False, and required has nothing in common with {"admin"}, so the last is True.

Next step

Check your understanding

9 quick questions

registered_students = {"Rohan", "Aditi", "Meera", "Dev"}submitted_assignment = {"Aditi", "Dev"} pending_students = registered_students - submitted_assignmentprint("Pending submission:", sorted(pending_students))