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Axiomatic Set Theory

Description: This quiz is designed to test your understanding of the fundamental concepts and axioms of Axiomatic Set Theory, a branch of mathematics that studies the properties of sets and their relationships.
Number of Questions: 14
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Tags: set theory axioms zermelo-fraenkel set theory russell's paradox
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Which of the following is an axiom of Zermelo-Fraenkel Set Theory?

  1. The Axiom of Extensionality

  2. The Axiom of Pairing

  3. The Axiom of Union

  4. The Axiom of Choice

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Correct Option:
Explanation:

The Axiom of Extensionality, the Axiom of Pairing, the Axiom of Union, and the Axiom of Choice are all axioms of Zermelo-Fraenkel Set Theory, which is a widely accepted foundation for modern mathematics.

What is the Axiom of Extensionality?

  1. Two sets are equal if and only if they have the same elements.

  2. A set can be defined by its properties.

  3. The union of two sets is the set of all elements that are in either set.

  4. Every set has a unique complement.

Show answer
Correct Option: A
Explanation:

The Axiom of Extensionality states that two sets are equal if and only if they have the same elements. This means that the order of the elements in a set does not matter, and that sets with the same elements are considered to be the same set.

What is the Axiom of Pairing?

  1. Two sets can be paired to form a new set.

  2. A set can be defined by its properties.

  3. The union of two sets is the set of all elements that are in either set.

  4. Every set has a unique complement.

Show answer
Correct Option: A
Explanation:

The Axiom of Pairing states that two sets can be paired to form a new set. This means that given two sets A and B, there exists a set C that contains all the ordered pairs (a, b) where a is an element of A and b is an element of B.

What is the Axiom of Union?

  1. Two sets can be paired to form a new set.

  2. A set can be defined by its properties.

  3. The union of two sets is the set of all elements that are in either set.

  4. Every set has a unique complement.

Show answer
Correct Option: C
Explanation:

The Axiom of Union states that the union of two sets A and B is the set of all elements that are in either A or B. This means that the union of two sets is the smallest set that contains all the elements of both sets.

What is the Axiom of Choice?

  1. Two sets can be paired to form a new set.

  2. A set can be defined by its properties.

  3. The union of two sets is the set of all elements that are in either set.

  4. Given a collection of non-empty sets, there exists a function that selects an element from each set.

Show answer
Correct Option: D
Explanation:

The Axiom of Choice states that given a collection of non-empty sets, there exists a function that selects an element from each set. This means that it is always possible to make a choice from a collection of sets, even if the collection is infinite.

What is Russell's Paradox?

  1. A set that contains itself as an element.

  2. A set that is empty.

  3. A set that is infinite.

  4. A set that is well-ordered.

Show answer
Correct Option: A
Explanation:

Russell's Paradox is a contradiction that arises from the idea of a set that contains itself as an element. This paradox led to the development of Axiomatic Set Theory, which is designed to avoid such contradictions.

What is the Zermelo-Fraenkel set theory?

  1. A set of axioms used to define the concept of a set.

  2. A set of axioms used to define the concept of a function.

  3. A set of axioms used to define the concept of a relation.

  4. A set of axioms used to define the concept of a number.

Show answer
Correct Option: A
Explanation:

The Zermelo-Fraenkel set theory is a set of axioms used to define the concept of a set. It is the most widely accepted foundation for modern mathematics.

Which of the following is not an axiom of Zermelo-Fraenkel set theory?

  1. The Axiom of Extensionality

  2. The Axiom of Pairing

  3. The Axiom of Union

  4. The Axiom of Infinity

Show answer
Correct Option: D
Explanation:

The Axiom of Infinity is not an axiom of Zermelo-Fraenkel set theory. It is an additional axiom that can be added to the theory to allow for the existence of infinite sets.

What is the Axiom of Regularity?

  1. Every non-empty set contains an element that is disjoint from the set.

  2. Every non-empty set contains an element that is a subset of the set.

  3. Every non-empty set contains an element that is a proper subset of the set.

  4. Every non-empty set contains an element that is a member of the set.

Show answer
Correct Option: A
Explanation:

The Axiom of Regularity states that every non-empty set contains an element that is disjoint from the set. This means that there is no set that contains itself as an element.

What is the Axiom of Replacement?

  1. If a set A is a subset of a set B, then there exists a set C that is isomorphic to A and a function f from B to C such that f(A) = C.

  2. If a set A is a subset of a set B, then there exists a set C that is isomorphic to A and a function f from C to B such that f(A) = C.

  3. If a set A is a subset of a set B, then there exists a set C that is isomorphic to A and a function f from A to C such that f(A) = C.

  4. If a set A is a subset of a set B, then there exists a set C that is isomorphic to A and a function f from B to A such that f(A) = C.

Show answer
Correct Option: A
Explanation:

The Axiom of Replacement states that if a set A is a subset of a set B, then there exists a set C that is isomorphic to A and a function f from B to C such that f(A) = C. This means that if we have a set A and a property P, then we can replace all the elements of A that satisfy P with a new element that also satisfies P, and the resulting set will be isomorphic to A.

What is the Power Set Axiom?

  1. For any set A, there exists a set B that contains all the subsets of A.

  2. For any set A, there exists a set B that contains all the elements of A.

  3. For any set A, there exists a set B that is isomorphic to A.

  4. For any set A, there exists a set B that is a proper subset of A.

Show answer
Correct Option: A
Explanation:

The Power Set Axiom states that for any set A, there exists a set B that contains all the subsets of A. This means that for any set A, we can create a new set that contains all the sets that can be formed by taking different combinations of the elements of A.

What is the Axiom of Choice?

  1. Given a collection of non-empty sets, there exists a function that selects an element from each set.

  2. Given a collection of non-empty sets, there exists a set that contains all the elements of all the sets in the collection.

  3. Given a collection of non-empty sets, there exists a set that is isomorphic to the union of all the sets in the collection.

  4. Given a collection of non-empty sets, there exists a set that is a proper subset of the union of all the sets in the collection.

Show answer
Correct Option: A
Explanation:

The Axiom of Choice states that given a collection of non-empty sets, there exists a function that selects an element from each set. This means that it is always possible to make a choice from a collection of sets, even if the collection is infinite.

What is the Axiom of Infinity?

  1. There exists a set that contains infinitely many elements.

  2. There exists a set that is isomorphic to the set of natural numbers.

  3. There exists a set that is a proper subset of the set of natural numbers.

  4. There exists a set that is a superset of the set of natural numbers.

Show answer
Correct Option: A
Explanation:

The Axiom of Infinity states that there exists a set that contains infinitely many elements. This means that there is a set that cannot be put into one-to-one correspondence with the set of natural numbers.

What is the Axiom of Foundation?

  1. Every non-empty set contains a minimal element.

  2. Every non-empty set contains a maximal element.

  3. Every non-empty set contains an element that is disjoint from the set.

  4. Every non-empty set contains an element that is a member of the set.

Show answer
Correct Option: A
Explanation:

The Axiom of Foundation states that every non-empty set contains a minimal element. This means that there is a set that does not contain any proper subsets.

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