Question

Which of the following sets is/are countable?

• A. The set of all functions from \( \{1, 2, 3, ...., 10\} \) to the set of all rational numbers
• B. The set of all functions from the set of all natural numbers to \( \{0, 1\} \)
• C. The set of all integer-valued sequences with only finitely many non-zero terms
• D. The set of all integer-valued sequences converging to 1

Hint:

- A set is countable if there is a one-to-one correspondence with the natural numbers.
- A set of all functions from a finite set to a countable set is countable.
- The set of all sequences with finitely many non-zero terms is countable.

Answer 1

The Correct Option is A

1) Understanding countability:
A set is countable if there exists a bijection between the set and the set of natural numbers. This means the set can either be finite or have the same cardinality as the natural numbers.
2) Analysis of the options:
(A) The set of all functions from \( \{1, 2, 3, ...., 10\} \) to the set of all rational numbers:
This set is countable. Since the domain is finite (with 10 elements), the number of functions from this set to the rationals is also countable. The set of rational numbers is countable, so the set of all functions is countable as well.
(B) The set of all functions from the set of all natural numbers to \( \{0, 1\} \):
This set is uncountable. The set of all functions from the natural numbers to a two-element set has the same cardinality as the set of all infinite sequences, which is uncountable.
(C) The set of all integer-valued sequences with only finitely many non-zero terms:
This set is countable. The set of sequences with only finitely many non-zero terms is a countable union of finite sets, and hence the set is countable.
(D) The set of all integer-valued sequences converging to 1:
This set is countable. The set of sequences converging to 1 can be viewed as a subset of a countable set of sequences, and hence is countable.
The correct answers are (A), (C), and (D).