A well-defined collection of distinct objects, called elements (or members). "Well-defined" means that for every object it is possible to determine whether it belongs to the collection or not.

An object that belongs to a set. We write x ∈ S to mean "x is an element of S" and x ∉ S to mean "x is not an element of S."

The set of all elements under consideration in a given context. Every set in the discussion is a subset of U. The choice of U depends on the problem.

ℕ = {0, 1, 2, 3, …} (natural numbers)

ℤ = {…, −2, −1, 0, 1, 2, …} (integers)

ℚ = the set of all rational numbers (fractions p/q where p, q ∈ ℤ and q ≠ 0)

ℝ = the set of all real numbers (rationals and irrationals)

A way of describing a set by explicitly listing all of its elements between curly braces, separated by commas. For example, {1, 2, 3} is the set containing exactly 1, 2, and 3.

A way of describing a set by specifying a property that its elements must satisfy. Written as {x ∈ S | P(x)}, which reads "the set of all x in S such that P(x) is true." The vertical bar | (or colon :) means "such that."

The set with no elements, denoted ∅ or {}. It is unique: there is exactly one set with no elements.

A set with exactly one element, written {a}. For example, {7} is the singleton containing only the number 7. Note that a ≠ {a} — an element is not the same as the singleton set containing it.

A is a subset of B, written A ⊆ B, if and only if every element of A is also an element of B. Formally: A ⊆ B ⟺ ∀x(x ∈ A → x ∈ B).

A is a proper subset of B, written A ⊂ B, if and only if A ⊆ B and A ≠ B. This means every element of A is in B, but B contains at least one element not in A.

Two sets A and B are equal, written A = B, if and only if A ⊆ B and B ⊆ A. This is called the principle of double containment (or mutual inclusion). In other words, A and B have exactly the same elements.

The union of A and B, written A ∪ B, is the set of all elements that belong to A or to B (or both). A ∪ B = {x | x ∈ A or x ∈ B}.

The intersection of A and B, written A ∩ B, is the set of all elements that belong to both A and B. A ∩ B = {x | x ∈ A and x ∈ B}.

The complement of A (with respect to the universal set U), written Ā, A', or Aᶜ, is the set of all elements in U that are not in A. Aᶜ = {x ∈ U | x ∉ A}.

The difference of A and B, written A \ B or A − B, is the set of all elements that belong to A but not to B. A \ B = {x | x ∈ A and x ∉ B}.

The symmetric difference of A and B, written A △ B, is the set of elements that belong to exactly one of A or B. Equivalently:

A △ B = (A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B).

To prove A ⊆ B, pick an arbitrary element from A and show it must also be in B.

Template:

1. "Let x ∈ A."

2. Use definitions to figure out properties of x.

3. Conclude "so x ∈ B."

4. Since x was arbitrary, A ⊆ B. □

This is called "element chasing" — you chase a generic element from one side to the other.

To prove A = B, show both directions:

1. Show A ⊆ B (every element of A is in B).

2. Show B ⊆ A (every element of B is in A).

This is called "double containment" or "mutual inclusion." If both sides contain each other, they must be the same set.

To prove A = ∅, show that no element can be in A. The most common way is proof by contradiction:

1. "Suppose for contradiction that x ∈ A."

2. Derive a contradiction (like x ∈ S and x ∉ S).

3. Conclude "no such x exists, so A = ∅."

The cardinality of a finite set A, written |A|, is the number of distinct elements in A. For example, |{a, b, c}| = 3.

The power set of A, written P(A) or 𝒫(A), is the set of all subsets of A, including the empty set ∅ and A itself. P(A) = {S | S ⊆ A}.

An ordered pair (a, b) is a pair of objects where the order matters. Two ordered pairs are equal if and only if their corresponding components are equal: (a, b) = (c, d) ⟺ a = c and b = d. In particular, (a, b) ≠ (b, a) unless a = b.

The Cartesian product of A and B, written A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. A × B = {(a, b) | a ∈ A and b ∈ B}.