Which of the following is/are true: The set of real

Definitions: Countable means a set can be put into a one-to-one correspondence with the natural numbers. Uncountable means it is not countable.

• The set of real numbers is countable: False. The real numbers are uncountable (Cantor's diagonal argument shows no complete listing of real numbers is possible).

• The set of rational numbers is countable: True. Rationals can be enumerated by arranging fractions as pairs of integers (numerator, denominator), ordering those pairs (for example by increasing sum of absolute values) and skipping duplicates, giving a sequence that lists every rational number.

• The set of integers is uncountable: False. The integers are countable; for example list them as 0, 1, -1, 2, -2, 3, -3, ... which gives a bijection with the natural numbers.

• The set of real numbers between 0 and 1/4 is uncountable: True. Any nontrivial interval of real numbers has the same cardinality as (0,1). A simple bijection x ↦ 4x maps (0,1/4) to (0,1), and (0,1) is uncountable by Cantor's diagonal argument.

Conclusion: The true statements are the countability of the rational numbers and the uncountability of the real numbers between 0 and 1/4. Therefore, the correct combination is: the statement that the rationals are countable and the statement that the real numbers between 0 and 1/4 are uncountable.