Cartesian products

An ordered pair (a, b) is two objects written in a fixed order: a is the first element, b the second. Order matters, just as in coordinate geometry: (1, 2) ≠ (2, 1). The two elements may even be the same, as in (3, 3). An ordered pair is not a set, which is why it uses round brackets.

Two ordered pairs are equal only when first equals first and second equals second. If (2x – 1, y + 4) = (5, 1), then 2x – 1 = 5 and y + 4 = 1, so x = 3, y = –3.

The Cartesian product A × B is the set of all ordered pairs whose first element comes from A and second from B: A × B = {(a, b) : a ∈ A, b ∈ B}.

• For A = {1, 2} and B = {p, q, r}: A × B = {(1, p), (1, q), (1, r), (2, p), (2, q), (2, r)}.
• Pair each element of A with every element of B, just like multiplying out brackets. So if n(A) = p and n(B) = q, then n(A × B) = pq: here 2 × 3 = 6.
• B × A starts with the elements of B: (p, 1), (p, 2), ... These are different pairs, so in general A × B ≠ B × A.
• If either set is empty, there is nothing to pair with: A × φ = φ.

Working backwards: A × B has 6 elements, and three of them are (1, p), (2, q) and (3, p). The first elements come from A and the second from B, so A ⊇ {1, 2, 3} and B ⊇ {p, q}. Since 3 × 2 = 6, A = {1, 2, 3} and B = {p, q}, and the missing pairs are (1, q), (2, p) and (3, q).

Three sets give ordered triplets: A × A × A = {(a, b, c) : a, b, c ∈ A}. For A = {0, 1} it has 2 × 2 × 2 = 8 triplets: (0, 0, 0), (0, 0, 1), ..., (1, 1, 1). Build it as (A × A) × A: take each pair from A × A and add each element of A on the end.

R × R is the set of all points of the plane, and R × R × R is the whole of three-dimensional space: every point is an ordered pair or triplet of real numbers.