Cardinality of a Product Set
Simple Explanation
The number of ordered pairs in A × B, written n(A × B), always equals n(A) × n(B) — the number of elements in A multiplied by the number of elements in B.
Why Do We Need It?
This counting shortcut means you never have to list out every pair just to know how many there are — a useful skill whenever you need to count combinations quickly.
Formula
Cardinality of a Product Set
n(A × B) = n(A) × n(B)
The number of ordered pairs in the product set A × B equals the number of elements in A times the number of elements in B.
- n(A × B)
- — the number of ordered pairs in the product set A × B
- n(A)
- — the number of elements in set A
- n(B)
- — the number of elements in set B
When to use it: Whenever you need to know how many ordered pairs a product set contains, without listing every pair by hand.
Worked Example
Count the elements of a product set
Set A has 5 elements and set B has 4 elements. How many elements does A × B have?
Why Does This Work?
Every one of the n(A) elements of A can be paired with each of the n(B) elements of B, independently — this is the same "independent choices multiply" counting principle used anywhere a first choice and a second choice combine freely.
Real-Life Example
Counting outfit combinations
A person has 6 shirts and 3 pairs of trousers and wants to know how many different outfits are possible.
Without listing every outfit, n(shirts × trousers) = 6 × 3 = 18 possible outfit combinations.
Practice
If n(A) = 7 and n(B) = 6, what is n(A × B)?
MediumCommon mistake
Adding the set sizes instead of multiplying them — n(A × B) = n(A) × n(B), not n(A) + n(B).
Quick Review
- n(A × B) = n(A) × n(B).
- This counts every ordered pair without needing to list them.
- It is the same counting principle behind counting combinations of independent choices.