A bijection between products of countable sets is a one-to-one correspondence between the elements of two countably infinite sets. This means that each element in one set is paired with exactly one element in the other set, and vice versa.
A bijection is a type of function that is both injective (one-to-one) and surjective (onto). This means that each element in the domain is paired with exactly one element in the co-domain, and every element in the co-domain has at least one corresponding element in the domain.
One example of a bijection between products of countable sets is the function f: ℕ x ℕ → ℕ, where f(m,n) = (2^m)(3^n). This function maps each pair of natural numbers to a unique natural number, creating a bijection between the product of two countably infinite sets (ℕ x ℕ) and a single countably infinite set (ℕ).
Bijections are commonly used in mathematics to prove that two sets have the same cardinality (size) by showing that there is a one-to-one correspondence between their elements. They are also used in combinatorics to count the number of ways that objects can be arranged or combined.
By establishing a bijection between two countably infinite sets, we can show that they have the same cardinality and are therefore "equally infinite". This has important implications in set theory and other areas of mathematics, as it allows us to compare the sizes of different infinite sets.