Subsets

A subset is one of the most fundamental ideas in set theory. We say that a set A is a subset of a set B if every single element that belongs to A also belongs to B. In other words, A is 'contained within' B — nothing in A is missing from B. For example, if A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, then A is a subset of B because 2, 4, and 6 all appear in B. The subset relationship is the primary way mathematicians compare sets by containment rather than by equality. It allows us to organise sets in a hierarchy, to reason about what properties are inherited, and to build powerful logical arguments. Every area of mathematics — from probability to calculus to abstract algebra — relies on subset notation and thinking. Two special cases arise immediately. First, the empty set ∅ is a subset of every set because there are no elements in ∅ that could fail to appear in another set. Second, every set is a subset of itself. These two facts might seem strange at first, but they follow directly from the definition and make the mathematics cleaner and more consistent.

Key takeaways
  • The Formal Definition
  • Proper Subsets vs Subsets
  • Proving A ⊆ B
  • The Power Set
  • Key Properties of ⊆

The Formal Definition

Set A is a subset of set B, written A ⊆ B, if and only if: for every element x, if x ∈ A then x ∈ B. In formal logic: A ⊆ B ⟺ (∀x)(x ∈ A → x ∈ B). This conditional statement is the key — we never claim anything about elements outside A.

Proper Subsets vs Subsets

A proper subset goes one step further. A is a proper subset of B, written A ⊂ B (or sometimes A ⊊ B), if A ⊆ B AND A ≠ B. This means B contains at least one element not in A. For example, {1, 3} ⊂ {1, 2, 3} is a proper subset, but {1, 2, 3} ⊆ {1, 2, 3} is a subset (not proper). The distinction matters in precise mathematical arguments.

Proving A ⊆ B

To prove A ⊆ B, you must show that an arbitrary element of A is also in B. The standard proof technique: 'Let x ∈ A. Then ... [reasoning] ... therefore x ∈ B.' This element-chasing method is universal. To disprove A ⊆ B, you only need one counterexample: find a single element x such that x ∈ A but x ∉ B.

The Power Set

The power set of a set A, written 𝒫(A) or 2^A, is the set of ALL subsets of A, including ∅ and A itself. If A = {a, b, c}, then 𝒫(A) = {∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}. If A has n elements, then 𝒫(A) has exactly 2^n elements. For n = 3 that gives 2³ = 8 subsets, which we can verify in this example.

Key Properties of ⊆

The subset relation satisfies three important properties on any collection of sets. (1) Reflexivity: A ⊆ A for every set A. (2) Antisymmetry: If A ⊆ B and B ⊆ A, then A = B — this is actually how mathematicians prove two sets are equal! (3) Transitivity: If A ⊆ B and B ⊆ C, then A ⊆ C. Together these make ⊆ a partial order on sets.

The Key Equation

A⊆B  ⟺  (∀x)(x∈A⇒x∈B)A \subseteq B \iff (\forall x)(x \in A \Rightarrow x \in B)
  • A, B — Sets (—): Any two sets being compared for containment.
  • \subseteq — Subset symbol (—): Read 'is a subset of' or 'is contained in'. Allows equality of the two sets.
  • \iff — If and only if (—): Biconditional; both sides must be simultaneously true or simultaneously false.
  • \forall x — Universal quantifier (—): Means 'for every element x' — the condition must hold for all elements, with no exceptions.
  • \Rightarrow — Implies (—): Logical conditional: if the left side is true, then the right side must also be true.

Worked Example

Problem. Let A = {x ∈ ℤ : x² < 10} and B = {−4, −3, −2, −1, 0, 1, 2, 3, 4}. Determine whether A ⊆ B, B ⊆ A, or A = B. Also find the power set of A ∩ B.

A⊂B,B⊈A,∣P(A∩B)∣=27=128A \subset B,\quad B \not\subseteq A,\quad |\mathcal{P}(A\cap B)| = 2^7 = 128

Answer. A ⊂ B (A is a proper subset of B); B is NOT a subset of A; A ≠ B. The power set of A ∩ B has 128 elements.

Figure
Diagram
Subset Relationships: A ⊆ B ⊆ UUBA2461357109A = {2, 4, 6}B = {1,2,3,4,5,6}U (universal)A ⊂ B ⊂ U (proper subsets shown by strictly smaller ellipses)
Origins · 1 minGeorg Cantor and how this was discovered
GC
Georg Cantor
1845–1918 · German

Georg Cantor invented modern set theory in the 1870s and 1880s, introducing the precise language of sets, subsets, and cardinality. He defined the subset relation formally, introduced power sets, and used them to prove the stunning result that the power set of any set is strictly larger than the set itself — even for infinite sets. His theorem |𝒫(A)| > |A| (for all sets A) is called Cantor's theorem, and it shows there is no largest infinity. Cantor's ideas were so radical that leading mathematicians of his era, including Leopold Kronecker, fiercely attacked them. Kronecker called Cantor a 'corrupter of youth.' Despite this, David Hilbert later declared: 'No one shall expel us from the paradise that Cantor has created.'

  • Introduction to Sets — Subsets are built on the basic definition of a set and set membership (∈), which must be understood first.
  • Union and Intersection of Sets — Union and intersection produce new sets from old ones; knowing whether these results are subsets of the originals is a core skill.
  • Power Sets — The power set is defined as the collection of all subsets of a given set, making it a direct extension of the subset concept.
  • Conditional Statements and Implication — The definition A ⊆ B is formally a universal conditional (∀x, x ∈ A ⇒ x ∈ B), so understanding logical implication is essential.
  • Venn and Euler Diagrams — Euler diagrams visually represent subset relationships, making them the primary tool for illustrating and reasoning about subsets.