Sets · step 4 of 13

The Cartesian product A × B

A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. Unlike sets, ordered pairs care about order: (1, 2) ≠ (2, 1), which is why A × B ≠ B × A in general. The cardinality rule |A × B| = |A| · |B| is just the product rule of counting, and it underpins the definition of relations and functions as sets of pairs.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Fix the orderThe first coordinate always comes from the first named set. Pair each element of A with every element of B, in a systematic sweep.
Count as a check|A × B| = |A| · |B|. If your list is shorter, you missed pairs.
Extend to more factorsA × B × C consists of ordered triples, with |A × B × C| = |A| · |B| · |C|. Note A × B × C and (A × B) × C are formally different but routinely identified.
Handle the empty caseIf either factor is empty, the product is empty: there is no pair to form.

Worked example

With A = {1, 2} and B = {x, y, z}, list A × B and give |B × A|.

|A| = 2 and |B| = 3, so |A × B| = 6.
Pair 1 with each of x, y, z: (1,x), (1,y), (1,z).
Pair 2 with each: (2,x), (2,y), (2,z).
|B × A| = 3 · 2 = 6 as well, but the sets differ — B × A contains (x,1), not (1,x).

Answer. A × B = {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)}; |B × A| = 6, though B × A ≠ A × B.

Where marks get dropped

These are the specific errors that cost credit on the cartesian product a × b questions — QED's rubric penalises each of them separately.

Writing pairs as sets, {a, b} instead of (a, b). Order is the whole point, and {a, b} = {b, a} destroys it.
Claiming A × B = B × A because the cardinalities match. Equal size does not mean equal set.
Forgetting that A × ∅ = ∅ regardless of how large A is.

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The Cartesian product A × B — frequently asked questions

Is A × B ever equal to B × A?

Only when A = B, or when one of them is empty. Otherwise some pair (a, b) with a ∉ B witnesses the difference.

What is A²?

Shorthand for A × A, the set of ordered pairs from A. Relations on A are exactly the subsets of A², which is why the notation is worth knowing.

Why is |A × B| = |A|·|B|?

By the product rule: choose the first coordinate |A| ways, then independently the second |B| ways. Every pair arises exactly once from such a choice.

The rest of Sets

Set-builder notation, operations, and set identities. Each subtopic below has its own method, worked example and mark-losing traps.

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