Final Answer:
(a) The set R×Q (Cartesian product of the real numbers and rational numbers) is uncountable.
(b) The set of irrational real numbers R\Q is uncountable.
(c) The set [0, ∞) is uncountable.
Step-by-step explanation:
(a) To prove that R×Q is uncountable, we can use a diagonalization argument. Assume for contradiction that R×Q is countable, meaning we can list its elements as (a_1, b_1), (a_2, b_2), and so on. Now, construct a real number x by taking the decimal expansion of a_i and changing the i-th digit to a digit different from b_i. This new real number (x, b_1) cannot be in the list since it differs from every element in at least one digit. This contradicts the assumption that R×Q is countable, proving it is uncountable.
(b) The set R\Q consists of irrational numbers. Using a similar diagonalization argument as in (a), we can show that R\Q is uncountable. Assume for contradiction that R\Q is countable, list its elements, and construct an irrational number that is not in the list, leading to a contradiction.
(c) The set [0, ∞) is uncountable and can be proved using Cantor's diagonal argument. Assume it is countable, list its elements, and construct a number not in the list by adding 1 to the diagonal element of each number. This new number is greater than any number in the list, contradicting the assumption that [0, ∞) is countable.