Set Definition and Notation

A set must be well-defined: every potential element either belongs or does not belong, with no ambiguity. For example, 'the set of all even integers between 1 and 10' is well-defined: {2, 4, 6, 8, 10}. By contrast, 'the set of tall students in class' is NOT well-defined because 'tall' is subjective. The objects inside a set are called elements or members.

In roster notation, all elements are written explicitly, separated by commas, inside curly braces: A = {2, 4, 6, 8, 10}. Order does not matter — {1, 2, 3} and {3, 1, 2} are the same set. Repetition is ignored — {1, 1, 2, 3} equals {1, 2, 3}. For very large or infinite sets, an ellipsis signals a continuing pattern: ℕ = {1, 2, 3, 4, ...} or {2, 4, 6, ..., 100}.

Set-builder notation has the form {variable | condition}, read as 'the set of all [variable] such that [condition]'. The vertical bar | (sometimes written as a colon :) means 'such that'. Example: {x ∈ ℤ | x² < 20} = {−4, −3, −2, −1, 0, 1, 2, 3, 4}. The expression before | tells you what kind of object you are describing; the expression after | gives the filter condition.

The symbol ∈ denotes membership: '3 ∈ {1, 2, 3, 4}' means '3 is an element of this set'. The symbol ∉ denotes non-membership: '5 ∉ {1, 2, 3, 4}'. These are among the most frequently used symbols in all of mathematics. Remember: ∈ connects an element to a set, not a set to a set.

Certain sets appear so often they have reserved symbols. The empty set ∅ (or {}) contains no elements. The natural numbers ℕ = {1, 2, 3, ...} (some definitions include 0). The integers ℤ = {..., −2, −1, 0, 1, 2, ...}. The rational numbers ℚ = {p/q | p, q ∈ ℤ, q ≠ 0}. The real numbers ℝ include all rationals and irrationals. The cardinality of a set A, written |A| or n(A), is the number of distinct elements it contains.

Two sets A and B are equal, written A = B, if and only if they contain exactly the same elements. Formally, A = B ⟺ (∀x, x ∈ A ⟺ x ∈ B). This means {1, 2, 3} = {3, 2, 1} = {1, 1, 2, 3} — order and repetition are irrelevant. To prove two sets are equal, you typically show each is a subset of the other.

• A — Set name (—): The label given to the set being defined.
• x — Variable (element placeholder) (—): A dummy variable representing a generic element being tested for membership.
• U — Universal set (domain) (—): The larger set from which elements are drawn, e.g., ℤ, ℝ, or a specific finite set.
• \mid — Such that (—): Separates the variable declaration from the condition; sometimes written as a colon ':'.
• P(x) — Predicate / property (—): A logical statement that is either true or false for each value of x; only elements for which P(x) is true are included.

Problem. Let A = {x ∈ ℤ | −3 ≤ x < 5} and B = {−3, −1, 1, 3}. (a) Write A in roster notation. (b) Determine whether B ⊆ A. (c) Find |A|.

Answer. A = {−3, −2, −1, 0, 1, 2, 3, 4}; B ⊆ A (B is a proper subset of A); |A| = 8.

• Subsets and Power Sets — Once you can define and represent sets, the next step is comparing them: a subset is a set whose every element belongs to another set, and the power set collects all possible subsets.
• Set Operations: Union, Intersection, and Complement — Set notation enables operations like union (A ∪ B), intersection (A ∩ B), and complement (Aᶜ), which are the arithmetic of sets and essential for probability and logic.
• Venn Diagrams — Venn diagrams are the primary visual tool for representing sets and their relationships; understanding set notation makes reading and constructing them far more precise.
• Universal and Existential Quantifiers — Set-builder notation uses logical predicates P(x); quantifiers ∀ (for all) and ∃ (there exists) are the formal logic tools that underpin those conditions.
• Domain and Range of Functions — The domain and range of a function are defined as sets, so fluency with set notation and set-builder form is essential for working with functions.