Which of the following set is uncountable

Concept. A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers, i.e. arranged in a single list (finite sets are also countable). A set is uncountable if no such listing exists — it is strictly “larger” than the natural numbers. The standard example of an uncountable set is the set of real numbers, proved uncountable by Cantor’s diagonal argument.

Applying this to each offered set:

• Natural numbers can be listed directly as 1, 2, 3, 4, …, which is itself a one-to-one correspondence with the natural numbers, so this set is countable.

• All integers can be listed as 0, 1, −1, 2, −2, 3, −3, …, alternating signs so every integer eventually appears at a fixed position, so this set is countable.

• Positive rationals p/q can be placed in an infinite grid (numerator × denominator) and traversed along successive diagonals, skipping repeated values; this produces a single list of all of them, so this set is countable.

Conclusion. All three named sets are countable, so the option asserting that none of the listed sets is uncountable is the one that holds. By contrast, the real numbers — had they been offered — would be the uncountable choice, since Cantor’s diagonal argument shows the reals in [0, 1] cannot be exhausted by any list.