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Countable and Uncountable Sets: Delving into the Infinite and Beyond

Description: Countable and Uncountable Sets: Delving into the Infinite and Beyond
Number of Questions: 15
Created by:
Tags: set theory countability uncountability infinite sets
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Which of the following sets is countable?

  1. The set of all even numbers

  2. The set of all prime numbers

  3. The set of all real numbers between 0 and 1

  4. The set of all subsets of the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all even numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all rational numbers

  2. The set of all irrational numbers

  3. The set of all real numbers

  4. The set of all complex numbers

Show answer
Correct Option: B
Explanation:

The set of all irrational numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets has the same cardinality as the set of natural numbers?

  1. The set of all integers

  2. The set of all rational numbers

  3. The set of all real numbers

  4. The set of all complex numbers

Show answer
Correct Option: A
Explanation:

The set of all integers has the same cardinality as the set of natural numbers because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets has a larger cardinality than the set of natural numbers?

  1. The set of all rational numbers

  2. The set of all real numbers

  3. The set of all complex numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: B
Explanation:

The set of all real numbers has a larger cardinality than the set of natural numbers because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: B
Explanation:

The set of all infinite subsets of the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is countable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all finite subsets of the natural numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: D
Explanation:

The set of all functions from the natural numbers to the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is countable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all finite subsets of the natural numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: B
Explanation:

The set of all infinite subsets of the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is countable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all finite subsets of the natural numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: D
Explanation:

The set of all functions from the natural numbers to the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is countable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all finite subsets of the natural numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: B
Explanation:

The set of all infinite subsets of the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is countable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: A
Explanation:

The set of all finite subsets of the natural numbers is countable because it can be put into a one-to-one correspondence with the set of natural numbers.

Which of the following sets is uncountable?

  1. The set of all finite subsets of the natural numbers

  2. The set of all infinite subsets of the natural numbers

  3. The set of all subsets of the natural numbers

  4. The set of all functions from the natural numbers to the natural numbers

Show answer
Correct Option: D
Explanation:

The set of all functions from the natural numbers to the natural numbers is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.

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