Cartesian Products: Ordered Pairs and Beyond

The Cartesian product creates ordered pairs from two sets.

If A = {1, 2} and B = {x, y}, then A × B = {(1,x), (1,y), (2,x), (2,y)}. Every element of A paired with every element of B. Four pairs in total.

The Cartesian product turns two sets into a set of ordered pairs.

That's the unlock. The coordinate plane is a Cartesian product: ℝ × ℝ pairs every real number with every other real number to create all points (x, y). Functions are subsets of Cartesian products. Relations are subsets of Cartesian products. The product is how set theory builds structure from ingredients.

The Definition

For sets A and B:

A × B = {(a, b) : a ∈ A and b ∈ B}

Every element of A paired with every element of B, in that order.

(a, b) is an ordered pair: the first component is from A, the second from B. Order matters.

Order Matters

(1, 2) ≠ (2, 1)

An ordered pair has a first element and a second element. Swapping them gives a different pair.

This is unlike sets, where {1, 2} = {2, 1}. The Cartesian product cares about position.

A Simple Example

Let A = {a, b} and B = {1, 2, 3}.

A × B = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}

Six pairs total: |A × B| = |A| × |B| = 2 × 3 = 6.

The Coordinate Plane

The real plane ℝ² is ℝ × ℝ:

ℝ × ℝ = {(x, y) : x ∈ ℝ and y ∈ ℝ}

Every point on the plane is an ordered pair of real numbers. René Descartes invented this — hence "Cartesian."

The x-axis is ℝ × {0}. The y-axis is {0} × ℝ. The origin is (0, 0).

Products Are Not Commutative

A × B ≠ B × A (unless A = B or one is empty)

{1, 2} × {a, b} = {(1,a), (1,b), (2,a), (2,b)} {a, b} × {1, 2} = {(a,1), (a,2), (b,1), (b,2)}

Different sets. The first has pairs like (1, a); the second has pairs like (a, 1).

Product with the Empty Set

A × ∅ = ∅ ∅ × B = ∅

If one set is empty, there are no elements to pair. No pairs can be formed.

Product with a Singleton

{a} × B = {(a, b) : b ∈ B}

This is effectively a "copy" of B, with each element tagged with a.

{} × {1, 2, 3} = {(, 1), (, 2), (, 3)}

Cardinality

|A × B| = |A| × |B|

If A has m elements and B has n elements, then A × B has m × n pairs.

This is why it's called the Cartesian product.

Triple Products and Beyond

A × B × C = {(a, b, c) : a ∈ A, b ∈ B, c ∈ C}

Ordered triples. Strictly speaking, this is (A × B) × C — pairs of (pairs and elements) — but we treat them as triples.

n-dimensional space: ℝⁿ = ℝ × ℝ × ... × ℝ (n times)

Points in n-dimensional space are n-tuples of real numbers.

Subsets of Cartesian Products

Important structures are subsets of Cartesian products:

Relations: A relation R from A to B is a subset of A × B. R = {(a, b) ∈ A × B : some condition on a and b}

Functions: A function f : A → B is a special relation where each a ∈ A appears in exactly one pair.

Graphs of functions: The graph of y = x² is {(x, x²) : x ∈ ℝ} ⊆ ℝ × ℝ.

Relations as Sets

The "less than" relation on {1, 2, 3}:

< = {(1,2), (1,3), (2,3)} ⊆ {1,2,3} × {1,2,3}

Each pair (a, b) in the relation satisfies a < b.

The "divisibility" relation on {1, 2, 3, 4, 6, 12}:

| = {(1,1), (1,2), (1,3), ..., (6,12), (12,12)}

(a, b) is in the relation if a divides b.

Functions as Sets

A function f : A → B can be defined as:

f = {(a, f(a)) : a ∈ A} ⊆ A × B

The function f(x) = x² on {-1, 0, 1, 2} is:

f = {(-1, 1), (0, 0), (1, 1), (2, 4)}

No two pairs share the same first component — that's what makes it a function.

Distributive Properties

A × (B ∪ C) = (A × B) ∪ (A × C) A × (B ∩ C) = (A × B) ∩ (A × C)

Cartesian product distributes over union and intersection.

Products and Subsets

If A ⊆ A' and B ⊆ B', then A × B ⊆ A' × B'.

Larger input sets give larger product sets.

Projections

Given a pair (a, b) ∈ A × B:

• The projection onto A is a.
• The projection onto B is b.

If S ⊆ A × B:

• π_A(S) = {a : (a, b) ∈ S for some b}
• π_B(S) = {b : (a, b) ∈ S for some a}

Projections "collapse" the pairs back to individual sets.

The Core Insight

The Cartesian product constructs compound objects from simple ones.

From two sets, you get pairs. From n sets, you get n-tuples. These tuples become coordinates, records, states, configurations — any time you need to bundle several pieces of information together.

Functions, relations, graphs, and spaces all live inside Cartesian products. The product is how set theory builds multi-dimensional structure from one-dimensional ingredients.

Part 8 of the Set Theory series.

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