To prove that the set {2, 4, 6, 8, 10, ...} is countable, we need to show that there exists a one-to-one correspondence between the set and the set of natural numbers, N = {1, 2, 3, 4, 5, ...}.
One way to establish such a correspondence is to define a function f: N → {2, 4, 6, 8, 10, ...} as follows:
f(n) = 2n
This function maps each natural number n to the corresponding even number 2n. Since every even number can be expressed in this form, the function f is onto.
To show that f is one-to-one, we can assume that f(m) = f(n) for some natural numbers m and n, and then show that m = n.
If f(m) = f(n), then 2m = 2n, which implies that m = n. Therefore, f is one-to-one.
Since we have shown that f is both onto and one-to-one, it follows that there exists a one-to-one correspondence between the set {2, 4, 6, 8, 10, ...} and the set of natural numbers, N. Therefore, the set {2, 4, 6, 8, 10, ...} is countable.