13.1 Basics of Sets

An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.

A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.

Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.

Note

Subset

A subset of a set $A$ is another set that contains only elements from the set $A$ , but may not contain all the elements of $A$ .

If $B$ is a subset of $A,$ we write $B ⊆ A$

A proper subset is a subset that is not identical to the original set – it contains fewer elements.

from itertools import combinations

# The book's three sets. EDIT any line and re-run.
A = {2, 4, 6, 8, 10, 12}      # a finite piece of "the set of all even numbers"
B = {2, 4, 6}
C = {2, 3, 4, 6}

for name, S in (("A", A), ("B", B), ("C", C)):
    print(f"{name} = {sorted(S)}     cardinality n({name}) = {len(S)}")

print("\nis-an-element-of  (the symbol in the book is a curly epsilon)")
for x in (2, 3, 7):
    print(f"   {x} in A ? {str(x in A):<5}   {x} in C ? {x in C}")

print("\nsubset and proper subset      (Python: S <= T is subset, S < T is proper)")
for n1, S, n2, T in (("B", B, "A", A), ("B", B, "C", C), ("C", C, "A", A)):
    print(f"   {n1} subset of {n2} ? {str(S <= T):<5}    proper ? {str(S < T):<5}")
    if not S <= T:
        extra = sorted(S - T)
        print(f"      not a subset because {extra} is in {n1} but not in {n2}")

print(f"\nevery subset of B = {sorted(B)}:")
b = sorted(B)
subsets = [set(c) for k in range(len(b) + 1) for c in combinations(b, k)]
for s in subsets:
    tag = "  <- not proper (it IS B)" if s == B else ""
    print("   { }" + tag if not s else f"   {sorted(s)}{tag}")
print(f"\nB has {len(subsets)} subsets = 2^{len(b)}; all but one of them are proper subsets.")
# Try it: change B to {2, 4, 6, 8} and watch the subset count double.

If $B$ is a proper subset of $A$ , we write $B ⊂ A$

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.