Sets & Frozensets

A set stores unique, unordered values. The key capability it adds over a list is O(1) membership testing and set algebra (union, intersection, difference).

primes  = {2, 3, 5, 7, 11}
letters = set("hello")          # {"h", "e", "l", "o"} — duplicates removed
empty   = set()                 # NOT {} — that's an empty dict

Core operations

s = {1, 2, 3, 4, 5}

# Membership — O(1)
3 in s          # True
9 in s          # False

# Add / remove
s.add(6)
s.discard(99)   # no error if missing
s.remove(1)     # KeyError if missing

# Size
len(s)          # 5

Set algebra

a = {1, 2, 3, 4}
b = {3, 4, 5, 6}

a | b           # union        → {1, 2, 3, 4, 5, 6}
a & b           # intersection → {3, 4}
a - b           # difference   → {1, 2}     (in a but not b)
a ^ b           # symmetric ∆  → {1, 2, 5, 6} (in one but not both)

# In-place versions
a |= b
a &= b

# Subset / superset
{1, 2} <= {1, 2, 3}   # True  — subset
{1, 2, 3} >= {1, 2}   # True  — superset
{1, 2}.isdisjoint({3, 4})  # True — no overlap

Set comprehensions

squares = {x**2 for x in range(10)}
# {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

`frozenset`

An immutable set. Since it is hashable, it can be used as a dictionary key or stored inside another set.

fs = frozenset({1, 2, 3})

# Can be a dict key
graph = {
    frozenset({"A", "B"}): 5,   # edge A–B with weight 5
    frozenset({"B", "C"}): 3,
}

# Set algebra works the same way
fs | frozenset({4, 5})     # frozenset({1, 2, 3, 4, 5})

Complexity at a glance

Operation	`set`	`list`
`x in s`	O(1)	O(n)
`add` / remove	O(1)	O(n)
union / inter.	O(n)	O(n²)

The fastest way to remove duplicates from a list (when order doesn’t matter):

data    = [3, 1, 4, 1, 5, 9, 2, 6, 5]
unique  = list(set(data))

# When order must be preserved, use dict.fromkeys (Python 3.7+):
unique_ordered = list(dict.fromkeys(data))
# [3, 1, 4, 5, 9, 2, 6]