Final answer:
To prove that the set of odd numbers has the same cardinality as the set of all natural numbers, one can establish a one-to-one correspondence (bijection) between the two sets. By defining a function that pairs each natural number with a unique odd number, we can demonstrate that both sets have the same number of elements.
Step-by-step explanation:
The cardinality of a set is a measure of the 'number of elements' in that set. To prove that the set of all odd numbers has the same cardinality as the set of all natural numbers, we can create a one-to-one correspondence between the two sets.
For every natural number 'n', there is exactly one odd number '2n-1'. This demonstrates that for every element in the set of natural numbers, there is a unique element in the set of odd numbers, proving that the two sets have equivalent cardinalities.
Formally, if we define the set of natural numbers as N = {1, 2, 3, ...} and the set of odd numbers as O = {1, 3, 5, ...}, we can establish the function f: N → O defined by f(n) = 2n - 1, which is a bijection. A bijection is both injective (no two elements in N map to the same element in O) and surjective (every element in O is mapped by some element in N).
Since such a bijection exists, N and O have the same cardinality, often denoted by the aleph-null (ℵ0), which symbolizes the cardinality of the set of all natural numbers or any set that can be put into a one-to-one correspondence with it.