* Cardinality of a set
[Definition] Sets A and B have the same cardinality (are equipotent) if there is a one-to-one mapping of A onto B.
(notated as |A| = |B|).
(this is bijection rule for counting)

* Cardinal number
There are sets called cardinal numbers with the property that (a) for every set X there is a unique cardinal |X| and (b) sets X and Y are equipotent iff |X| is equals to |Y|.

* Cardinality of a union
|A ∪ B| = |A| + |B| if A and B are disjoint
(sum rule for counting)
proof: if |A| = a and |B| = b, i.e., there are bijections f: a -> A and g: b -> B, then f ∪ g is a bijection from a+b to A ∪ B.

* Cardinality of a cartesian product
|A × B| = |A| × |B|
(product rule for counting)
proof: similar to sum rule.