"Denumerable" refers to a set that can be counted or put in a one-to-one correspondence with the natural numbers (1, 2, 3, ...). This means that each element in the set can be associated with a unique natural number.
To prove that two sets, A and B, are denumerable, you must show that there exists a one-to-one correspondence between the elements of A and B. This means that each element in A must be paired with a unique element in B, and vice versa.
An example of a denumerable set is the set of positive even numbers. Each even number can be paired with a unique natural number (e.g. 2 with 1, 4 with 2, 6 with 3, etc.), making it denumerable.
If AUB is denumerable, it means that there exists a one-to-one correspondence between the elements of AUB and the natural numbers. This implies that both A and B are denumerable because A and B are subsets of AUB, and any subset of a denumerable set is also denumerable.
Yes, there are other ways to prove that AUB is denumerable. One way is to show that AUB is countable, meaning that the elements of AUB can be listed in a sequence. Another way is to show that AUB is infinite, meaning that it has an infinite number of elements. Both of these properties imply that AUB is denumerable.