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Cardinality and Set Operations: A Journey into the World of Infinite Sets

Description: Welcome to the quiz on Cardinality and Set Operations, where we'll explore the fascinating world of infinite sets and their properties. Get ready to test your understanding of set theory concepts and operations.
Number of Questions: 15
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Tags: set theory cardinality infinite sets set operations
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Which of the following sets has a cardinality of (\aleph_0)?

  1. The set of all natural numbers (\mathbb{N})

  2. The set of all real numbers (\mathbb{R})

  3. The set of all even integers (2\mathbb{Z})

  4. The set of all prime numbers (\mathbb{P})

Show answer
Correct Option: A
Explanation:

The set of all natural numbers (\mathbb{N}) has a cardinality of (\aleph_0) because it can be put into one-to-one correspondence with the set of all integers (\mathbb{Z}).

Which of the following sets is uncountable?

  1. The set of all rational numbers (\mathbb{Q})

  2. The set of all algebraic numbers (\mathbb{A})

  3. The set of all transcendental numbers (\mathbb{T})

  4. The set of all constructible numbers (\mathbb{C})

Show answer
Correct Option: C
Explanation:

The set of all transcendental numbers (\mathbb{T}) is uncountable, meaning it cannot be put into one-to-one correspondence with the set of all natural numbers (\mathbb{N}).

What is the cardinality of the power set of a set with (n) elements?

  1. (n)

  2. (2^n)

  3. (n^2)

  4. (n!)

Show answer
Correct Option: B
Explanation:

The cardinality of the power set of a set with (n) elements is (2^n) because each element of the power set is a subset of the original set, and there are (2^n) possible subsets.

Which of the following operations is not associative on sets?

  1. Union (\cup)

  2. Intersection (\cap)

  3. Symmetric difference (\Delta)

  4. Complement ())

Show answer
Correct Option: D
Explanation:

The complement operation ()) is not associative on sets because ((A))^c (\ne) ((A\cup B)^c) in general.

What is the distributive law of set operations?

  1. (A\cap(B\cup C) = (A\cap B)\cup(A\cap C))

  2. (A\cup(B\cap C) = (A\cup B)\cap(A\cup C))

  3. (A\Delta(B\cup C) = (A\Delta B)\cup(A\Delta C))

  4. (A\Delta(B\cap C) = (A\Delta B)\cap(A\Delta C))

Show answer
Correct Option: B
Explanation:

The distributive law of set operations states that (A\cup(B\cap C) = (A\cup B)\cap(A\cup C)).

Which of the following is an example of a bijection between two sets?

  1. The function (f(x) = x^2) from (\mathbb{R}) to (\mathbb{R})

  2. The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1])

  3. The function (f(x) = \lfloor x \rfloor) from (\mathbb{R}) to (\mathbb{Z})

  4. The function (f(x) = \lceil x \rceil) from (\mathbb{R}) to (\mathbb{Z})

Show answer
Correct Option: B
Explanation:

The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1]) is a bijection because it is both one-to-one and onto.

What is the cardinality of the set of all subsets of a set with (n) elements?

  1. (n)

  2. (2^n)

  3. (n^2)

  4. (n!)

Show answer
Correct Option: B
Explanation:

The cardinality of the set of all subsets of a set with (n) elements is (2^n) because each element of the power set is a subset of the original set, and there are (2^n) possible subsets.

Which of the following sets is not closed under the operation of union?

  1. The set of all natural numbers (\mathbb{N})

  2. The set of all even integers (2\mathbb{Z})

  3. The set of all rational numbers (\mathbb{Q})

  4. The set of all real numbers (\mathbb{R})

Show answer
Correct Option: B
Explanation:

The set of all even integers (2\mathbb{Z}) is not closed under the operation of union because the union of two even integers is not necessarily even.

What is the cardinality of the set of all functions from a set with (m) elements to a set with (n) elements?

  1. (m)

  2. (n)

  3. (m^n)

  4. (n^m)

Show answer
Correct Option: D
Explanation:

The cardinality of the set of all functions from a set with (m) elements to a set with (n) elements is (n^m) because there are (n) choices for the image of each element in the domain.

Which of the following is an example of a one-to-one function?

  1. The function (f(x) = x^2) from (\mathbb{R}) to (\mathbb{R})

  2. The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1])

  3. The function (f(x) = \lfloor x \rfloor) from (\mathbb{R}) to (\mathbb{Z})

  4. The function (f(x) = \lceil x \rceil) from (\mathbb{R}) to (\mathbb{Z})

Show answer
Correct Option: B
Explanation:

The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1]) is one-to-one because each element in the domain is mapped to a unique element in the range.

What is the cardinality of the set of all real numbers between 0 and 1?

  1. (\aleph_0)

  2. (\aleph_1)

  3. (\continuum)

  4. (\infty)

Show answer
Correct Option: C
Explanation:

The cardinality of the set of all real numbers between 0 and 1 is (\continuum), which is equal to the cardinality of the set of all real numbers.

Which of the following is an example of an onto function?

  1. The function (f(x) = x^2) from (\mathbb{R}) to (\mathbb{R})

  2. The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1])

  3. The function (f(x) = \lfloor x \rfloor) from (\mathbb{R}) to (\mathbb{Z})

  4. The function (f(x) = \lceil x \rceil) from (\mathbb{R}) to (\mathbb{Z})

Show answer
Correct Option: C
Explanation:

The function (f(x) = \lfloor x \rfloor) from (\mathbb{R}) to (\mathbb{Z}) is onto because every element in the range is the image of at least one element in the domain.

What is the cardinality of the set of all subsets of a set with (\aleph_0) elements?

  1. (\aleph_0)

  2. (\aleph_1)

  3. (\continuum)

  4. (\infty)

Show answer
Correct Option: C
Explanation:

The cardinality of the set of all subsets of a set with (\aleph_0) elements is (\continuum), which is equal to the cardinality of the set of all real numbers.

Which of the following is an example of a bijective function?

  1. The function (f(x) = x^2) from (\mathbb{R}) to (\mathbb{R})

  2. The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1])

  3. The function (f(x) = \lfloor x \rfloor) from (\mathbb{R}) to (\mathbb{Z})

  4. The function (f(x) = \lceil x \rceil) from (\mathbb{R}) to (\mathbb{Z})

Show answer
Correct Option: B
Explanation:

The function (f(x) = \sin(x)) from (\mathbb{R}) to ([0, 1]) is bijective because it is both one-to-one and onto.

What is the cardinality of the set of all functions from a set with (\aleph_0) elements to a set with (\aleph_1) elements?

  1. (\aleph_0)

  2. (\aleph_1)

  3. (\continuum)

  4. (\infty)

Show answer
Correct Option: C
Explanation:

The cardinality of the set of all functions from a set with (\aleph_0) elements to a set with (\aleph_1) elements is (\continuum), which is equal to the cardinality of the set of all real numbers.

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