Lesson 2 of 10 · 9 min
Working with Cartesian products
NCERT §2.2
The shop's stock list writes each shirt as (size, colour): (M, navy), (L, white) and so on. It still has 8 entries. Is Z × K the same set as K × Z?
The lesson in notes
In short
Order matters: A × B and B × A contain different pairs in general, so A × B ≠ B × A, although n(A × B) = n(B × A). For P = {a, b} and Q = {5}, P × Q = {(a, 5), (b, 5)} while Q × P = {(5, a), (5, b)}.
For non-empty A and B, A × B = B × A holds only when A = B.
If A and B are non-empty and at least one of them is infinite, then A × B is infinite; for example N × {1} has the infinitely many pairs (1, 1), (2, 1), (3, 1), ….
The product distributes over intersection and union: A × (B ∩ C) = (A × B) ∩ (A × C) and A × (B ∪ C) = (A × B) ∪ (A × C). With A = {1, 2}, B = {2, 3} and C = {3, 4}, both sides of the first identity equal {(1, 3), (2, 3)}.
The factors can be read back from the product: A is the set of first entries of A × B and B the set of second entries. If A × A has 4 elements and contains (2, 5), then 2, 5 ∈ A and n(A) = 2, so A = {2, 5} and A × A = {(2, 2), (2, 5), (5, 2), (5, 5)}.
When n(A × B) is given together with a few pairs, use n(A × B) = n(A) × n(B) to fix how many elements each factor has before listing them.