1. Show that the union of two disjoint countable sets is also countable.

2. Find the error in the following proof that the set of binary numbers is uncountable.

Let B be the set of binary numbers. We will prove that B is uncountable by using Cantor's diagonalization argument.

1. Assume that B is countable and a correspondence f:N → B exists:

                 n     f(n)
               ------------
                 1      0
                 2      1
                 3     10
                 4     11
                 ...   ...

2. Can we find a new binary number that is different from every binary number in the above list?
Yes. Let's call the new number h. Construct h such that its first digit is the reverse of the first digit of f(1), its second digit is the reverse of the second digit of f(2), and so on. In other words, if subscript i denotes the i^th digit from the left:

h_i = 1 - f_i(i).

3. We ensure that there is no a ∈ N such that f(a) = h. Thus f is not onto. This is a contradiction. Therefore B is not countable.

3. Let B be the set of total functions from N to N. Show that B is uncountable, using a proof by diagonalization.

4. Let B be the set of monotone-increasing total functions from N to N. Show that B is uncountable, using a proof by diagonalization.

Note: A total function f from N to N is monotone-increasing if f(n) < f(n+1) for all n ∈ N.