Set Notation: The Symbols That Define Collections
Mathematics invented symbols so you don't have to keep writing "is a member of" and "the set of all x such that."
Once you learn six symbols, you can read any set expression. That's it. Six symbols unlock the entire notational system.
∈ means "is an element of" ∉ means "is not an element of" { } means "the set containing..." : or | means "such that" ⊆ means "is a subset of" ∅ means "the empty set"
That's the unlock. Set notation looks intimidating until you realize it's just shorthand. Every symbol replaces a phrase you'd otherwise have to write out.
The Element Symbol: ∈
3 ∈ {1, 2, 3, 4, 5}
This says "3 is an element of the set {1, 2, 3, 4, 5}." The element is on the left; the set is on the right.
Think of ∈ as a stylized lowercase "e" for "element."
7 ∉ {1, 2, 3, 4, 5}
The slash negates it: "7 is not an element of the set {1, 2, 3, 4, 5}."
Curly Braces: { }
Curly braces define a set by listing its elements.
{a, b, c} — the set containing a, b, and c. {1, 2, 3, ..., 100} — the integers from 1 to 100 (the ellipsis indicates the pattern continues). {} or ∅ — the empty set.
Braces are the container. Whatever's inside the braces are the elements.
Set-Builder Notation
When you can't list all elements, describe them:
{x : x is a positive even integer}
Read this as: "The set of all x such that x is a positive even integer."
The colon (or sometimes a vertical bar |) separates the variable from the condition. Left of the colon: what you're collecting. Right of the colon: what property it must have.
More examples:
{x ∈ ℝ : x² < 9} — all real numbers whose square is less than 9 (that's -3 < x < 3)
{n ∈ ℤ : n > 0 and n divides 12} — positive integers that divide 12, which is {1, 2, 3, 4, 6, 12}
{(x, y) : x² + y² = 1} — all points on the unit circle
Standard Number Sets
Mathematicians use special letters for common sets:
ℕ — Natural numbers: {0, 1, 2, 3, ...} or {1, 2, 3, ...} (conventions vary) ℤ — Integers: {..., -2, -1, 0, 1, 2, ...} (from German Zahlen, "numbers") ℚ — Rational numbers: fractions p/q where p, q ∈ ℤ and q ≠ 0 (from quotient) ℝ — Real numbers: all points on the number line ℂ — Complex numbers: a + bi where a, b ∈ ℝ
These are the building blocks. You'll see expressions like "Let x ∈ ℝ" constantly.
Subset Notation: ⊆ and ⊂
A ⊆ B means "A is a subset of B" — every element of A is also in B.
{1, 2} ⊆ {1, 2, 3, 4} ✓ {1, 5} ⊆ {1, 2, 3, 4} ✗ (because 5 ∉ {1, 2, 3, 4})
Some authors use ⊂ for "proper subset" (subset but not equal) and ⊆ for "subset or equal." Others use ⊂ the way we use ⊆ here. When precision matters, authors define their convention.
Superset Notation: ⊇
A ⊇ B means "A is a superset of B" — A contains everything B contains.
It's just ⊆ reversed: A ⊇ B means B ⊆ A.
{1, 2, 3, 4} ⊇ {1, 2}
The Empty Set: ∅
∅ is the set with no elements. It's also written {}.
∅ ⊆ A for any set A. (Vacuously true: there's no element of ∅ that fails to be in A.)
∅ ≠ {∅}. The first has zero elements. The second has one element (the empty set itself).
Set Equality
A = B means A and B have exactly the same elements.
To prove A = B, you typically show:
- A ⊆ B (every element of A is in B)
- B ⊆ A (every element of B is in A)
If both hold, the sets are identical.
Union: ∪
A ∪ B is the set of elements in A or B (or both).
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
Think of ∪ as a "cup" that collects from both sides.
Intersection: ∩
A ∩ B is the set of elements in A and B (in both).
{1, 2, 3} ∩ {3, 4, 5} = {3}
Think of ∩ as a "cap" — only what's covered by both.
Set Difference: \ or -
A \ B (or A - B) is the set of elements in A but not in B.
{1, 2, 3, 4, 5} \ {3, 4} = {1, 2, 5}
You're subtracting B's elements from A.
Complement: Aᶜ or A'
Aᶜ is everything not in A (relative to some universal set U).
If U = {1, 2, 3, 4, 5} and A = {1, 2}, then Aᶜ = {3, 4, 5}.
The complement depends on context — you need to know what "everything" means.
Cardinality: |A|
|A| denotes the cardinality of A — the number of elements.
|{a, b, c}| = 3 |∅| = 0 |ℕ| = ℵ₀ (infinity, specifically "countable infinity")
For finite sets, cardinality is just the count. For infinite sets, it gets more interesting.
Power Set: 𝒫(A)
𝒫(A) is the set of all subsets of A.
If A = {1, 2}, then 𝒫(A) = {∅, {1}, {2}, {1, 2}}.
A set with n elements has 2ⁿ subsets. Why? Each element is either in a subset or not — two choices per element.
Reading Complex Expressions
{x ∈ ℝ : x² - 5x + 6 = 0}
Translation: "The set of all real numbers x such that x² - 5x + 6 = 0."
That equation factors as (x-2)(x-3) = 0, so the set is {2, 3}.
∀x ∈ A, P(x)
"For all x in A, property P holds." (The ∀ is a universal quantifier — we'll cover this in logic.)
The Core Insight
Set notation is a compression algorithm for mathematical English.
Every symbol has a direct translation. Once you internalize the six core symbols — ∈, ∉, { }, :, ⊆, ∅ — plus the operations ∪, ∩, \, and ᶜ, you can read any set expression by expanding it into words.
The notation isn't meant to obscure. It's meant to make complex statements short enough to reason about. A proof involving "the set of all x in A such that x is also in B" is clearer as "A ∩ B."
Part 2 of the Set Theory series.
Previous: What Is Set Theory? The Language Mathematics Speaks to Itself Next: Subsets and Supersets: When One Set Lives Inside Another
Comments ()