What is constructive mathematics?

It is the continuity of a function such as a sin x proven as a constructive bound on the modulus of continuity

It is the stagnation of a function such as a sin x proven as a constructive bound on the modulus of continuity

It is the improvement of a function such as a sin x proven as a constructive bound on the modulus of continuity

It is the continuity of a function such as a tan x proven as a constructive bound on the modulus of continuity

What is the axiom of infinity?

It is 1 of the axioms of Zermelo-Fraenkel set theory.

It's the principle behind plus and minus infinity.

It is 1 of the axioms of the Newton set theory.

What is an existential quantification?

It's a type of quantifier, a logical constant which is interpreted as "there exists"; "there is at least one", etc.

It's a type of function, which is interpreted as "there exists"; "there is at least one", etc.

It's what led to the discovery of the number "0"

It's what helps us give value to numbers.

What is a big O notation?

It's a mathematical notation describing the limiting behavior of a function when the argument tends towards a particular value of infinity.

It's a theorem which defines the circumference of earth.

It is a rule which defines the circumference of earth.

It is the great rule that helped navigators during the 15th Century.

It's the trigonometric function of an angle.

It's the third side of a triangle.

What is the law of excluded middle?

It's a law which states that for any proposition, either that a proposition is true or that its negation is true.

It's the portion of solution which includes number from 0 to + infinity.

What is an axiom of choice?

It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is empty.

It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is solvable.

It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is superior to 1.

What is a first-order logic?

It's a collection of formal systems used in math, philosophy, linguistics, and computer science.

It's a collection of formal systems used in math, philosophy, and computer science.

It's a collection of formal systems used in math and computer science.

It's a collection of formal systems used in math, philosophy, and computer science.

What is a numerical analysis?

It is a study of algorithms that use numerical approximation for the problems of mathematical analysis.

It is a study of theorems that use numerical approximation for the problems of mathematical analysis.

It is a study of algorithms that use statistical approximation for the problems of mathematical analysis.

It is a study of algorithms that use symbols for the problems of mathematical analysis.

Test Your Knowledge About Existence Theorems

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Test Your Knowledge About Existence Theorems - Quiz

Existence theorems regroup any mathematical or scientific statement beginning with "there exists. . .", or "for all x,y there exists. . .", etc. In other words, it is a theorem with a prenex normal form involving the existential quantifier. Do you want to take a chance and pass our quiz about this type of theorem? Try it and see how well you do.

Questions and Answers

1.

What is constructive mathematics?
- A.
  
  It is the continuity of a function such as a sin x proven as a constructive bound on the modulus of continuity
- B.
  
  It is the stagnation of a function such as a sin x proven as a constructive bound on the modulus of continuity
- C.
  
  It is the improvement of a function such as a sin x proven as a constructive bound on the modulus of continuity
- D.
  
  It is the continuity of a function such as a tan x proven as a constructive bound on the modulus of continuity
Correct Answer
A. It is the continuity of a function such as a sin x proven as a constructive bound on the modulus of continuity
2.

What is the axiom of infinity?
- A.
  
  It's a very complicated rule.
- B.
  
  It's the principle behind plus and minus infinity.
- C.
  
  It is 1 of the axioms of Zermelo-Fraenkel set theory.
- D.
  
  It is 1 of the axioms of the Newton set theory.
Correct Answer
C. It is 1 of the axioms of Zermelo-Fraenkel set theory.

Explanation
The axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. This axiom states that there exists at least one set, denoted as "omega," that contains the empty set and is closed under the successor function. In other words, it guarantees the existence of an infinite set in the theory. This axiom is fundamental in establishing the existence of infinite sets and is widely accepted in modern set theory.

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3.

What is an existential quantification?
- A.
  
  It's a type of function, which is interpreted as "there exists"; "there is at least one", etc.
- B.
  
  It's what led to the discovery of the number "0"
- C.
  
  It's what helps us give value to numbers.
- D.
  
  It's a type of quantifier, a logical constant which is interpreted as "there exists"; "there is at least one", etc.
Correct Answer
D. It's a type of quantifier, a logical constant which is interpreted as "there exists"; "there is at least one", etc.

Explanation
The correct answer explains that existential quantification is a type of quantifier in logic. It is a logical constant that is interpreted as "there exists" or "there is at least one". This means that it is used to express the existence of at least one object that satisfies a given condition.

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4.

What is a big O notation?
- A.
  
  It's a mathematical notation describing the limiting behavior of a function when the argument tends towards a particular value of infinity.
- B.
  
  It's a theorem which defines the circumference of earth.
- C.
  
  It is a rule which defines the circumference of earth.
- D.
  
  It is the great rule that helped navigators during the 15th Century.
Correct Answer
A. It's a mathematical notation describing the limiting behavior of a function when the argument tends towards a particular value of infinity.
5.

What is a sine?
- A.
  
  It's the trigonometric function of an angle.
- B.
  
  It's the third side of a triangle.
- C.
  
  It's the degree of an angle.
- D.
  
  It's a rule in geometry.
Correct Answer
A. It's the trigonometric function of an angle.

Explanation
The correct answer is "It's the trigonometric function of an angle." In trigonometry, a sine is a mathematical function that relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. It is commonly denoted as sin and is used to calculate angles and sides in various mathematical and scientific applications.

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6.

What is the law of excluded middle?
- A.
  
  It's the portion of solution which includes number from 0 to + infinity.
- B.
  
  It's a law which states that for any proposition, either that a proposition is true or that its negation is true.
- C.
  
  It's the perfect half.
- D.
  
  It's the median of a function.
Correct Answer
B. It's a law which states that for any proposition, either that a proposition is true or that its negation is true.

Explanation
The law of excluded middle states that for any proposition, it is either true or its negation is true. This means that there is no middle ground or alternative possibility - a proposition cannot be both true and not true at the same time. This principle is fundamental in classical logic and is often used as a basis for reasoning and proof. It allows for clear and binary distinctions between truth values, simplifying logical analysis and decision-making processes.

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7.

What is an axiom of choice?
- A.
  
  It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is empty.
- B.
  
  It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
- C.
  
  It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is solvable.
- D.
  
  It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is superior to 1.
Correct Answer
B. It is an axiom of set of theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

Explanation
The correct answer is that the axiom of choice is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. This means that for any collection of sets, each containing at least one element, it is possible to choose one element from each set to form a new set. This axiom is an important tool in mathematics and has applications in various areas such as analysis, topology, and algebra. It allows for the construction of certain mathematical objects that would otherwise be impossible to define.

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8.

How is an axiom of choice also called?
- A.
  
  CA
- B.
  
  AC.
- C.
  
  AB
- D.
  
  AX
Correct Answer
B. AC.

Explanation
An axiom of choice is also known as AC.

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9.

What is a first-order logic?
- A.
  
  It's a collection of formal systems used in math, philosophy, linguistics, and computer science.
- B.
  
  It's a collection of formal systems used in math, philosophy, and computer science.
- C.
  
  It's a collection of formal systems used in math and computer science.
- D.
  
  It's a collection of formal systems used in math, philosophy, and computer science.
Correct Answer
A. It's a collection of formal systems used in math, pHilosopHy, linguistics, and computer science.

Explanation
The correct answer is "It's a collection of formal systems used in math, philosophy, linguistics, and computer science." This answer encompasses all the disciplines mentioned in the question and accurately describes the scope of first-order logic.

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10.

What is a numerical analysis?
- A.
  
  It is a study of algorithms that use numerical approximation for the problems of mathematical analysis.
- B.
  
  It is a study of theorems that use numerical approximation for the problems of mathematical analysis.
- C.
  
  It is a study of algorithms that use statistical approximation for the problems of mathematical analysis.
- D.
  
  It is a study of algorithms that use symbols for the problems of mathematical analysis.
Correct Answer
A. It is a study of algorithms that use numerical approximation for the problems of mathematical analysis.

Explanation
Numerical analysis is a field of study that focuses on developing and analyzing algorithms that involve numerical approximation to solve problems in mathematical analysis. This involves using numerical methods and techniques to approximate solutions to mathematical problems, such as finding roots of equations, solving differential equations, and evaluating integrals. By using numerical approximation, these algorithms provide efficient and practical solutions to complex mathematical problems that may not have exact analytical solutions.

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