Consider { U = { X \mid X $}$ Is A Positive Integer Greater Than 1 }.Which Is An Empty Set?A. { { X \mid X \in U \text{ And } \frac{1}{2}x \text{ Is Prime} }$} B . \[ B. \[ B . \[ { X \mid X \in U \text{ And } 2x \text{ Is Prime}

Introduction

In mathematics, an empty set is a set that contains no elements. It is denoted by the symbol ∅ and is often referred to as the "null set." Empty sets are an essential concept in set theory, and understanding them is crucial for solving various mathematical problems. In this article, we will explore the concept of empty sets and examine two sets, A and B, to determine which one is empty.

What is a Set?

A set is a collection of unique objects, known as elements or members, that can be anything: numbers, letters, people, or even other sets. Sets are often represented using curly brackets {} and are denoted by a capital letter, such as U. The elements of a set are listed inside the curly brackets, separated by commas.

What is an Empty Set?

An empty set is a set that contains no elements. It is denoted by the symbol ∅ and is often referred to as the "null set." An empty set is a set that has no elements, and it is not the same as a set with no elements. A set with no elements is a set that has been defined but has not been populated with any elements.

Set A: { x ∣ x ∈ U and 1/2x is prime }

Set A is defined as the set of all elements x in U such that 1/2x is prime. To determine if this set is empty, we need to examine the elements of U and check if 1/2x is prime for each element.

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.

Is 1/2x Prime?

To determine if 1/2x is prime, we need to examine the elements of U and check if 1/2x is a prime number. Since U is the set of all positive integers greater than 1, we can start by examining the smallest element of U, which is 2.

2/2 = 1

When we divide 2 by 2, we get 1, which is not a prime number. Therefore, 1/2x is not prime for x = 2.

3/2 = 1.5

When we divide 3 by 2, we get 1.5, which is not a prime number. Therefore, 1/2x is not prime for x = 3.

5/2 = 2.5

When we divide 5 by 2, we get 2.5, which is not a prime number. Therefore, 1/2x is not prime for x = 5.

7/2 = 3.5

When we divide 7 by 2, we get 3.5, which is not a prime number. Therefore, 1/2x is not prime for x = 7.

Conclusion for Set A

Based on our examination of the elements of U, we can conclude that 1/2x is not prime for any element x in U. Therefore, set A is empty.

Set B: { x ∣ x ∈ U and 2x is prime }

Set B is defined as the set of all elements x in U such that 2x is prime. To determine if this set is empty, we need to examine the elements of U and check if 2x is prime for each element.

Is 2x Prime?

To determine if 2x is prime, we need to examine the elements of U and check if 2x is a prime number. Since U is the set of all positive integers greater than 1, we can start by examining the smallest element of U, which is 2.

2 × 2 = 4

When we multiply 2 by 2, we get 4, which is not a prime number. Therefore, 2x is not prime for x = 2.

3 × 2 = 6

When we multiply 3 by 2, we get 6, which is not a prime number. Therefore, 2x is not prime for x = 3.

5 × 2 = 10

When we multiply 5 by 2, we get 10, which is not a prime number. Therefore, 2x is not prime for x = 5.

7 × 2 = 14

When we multiply 7 by 2, we get 14, which is not a prime number. Therefore, 2x is not prime for x = 7.

Conclusion for Set B

Based on our examination of the elements of U, we can conclude that 2x is not prime for any element x in U. Therefore, set B is empty.

Conclusion

Q: What is an empty set?

A: An empty set is a set that contains no elements. It is denoted by the symbol ∅ and is often referred to as the "null set."

Q: How do you represent an empty set?

A: An empty set is represented by the symbol ∅, which is a special symbol used to denote a set with no elements.

Q: What is the difference between an empty set and a set with no elements?

A: An empty set is a set that has been defined but has no elements, whereas a set with no elements is a set that has not been populated with any elements.

Q: Can an empty set have elements added to it?

A: Yes, an empty set can have elements added to it. In fact, an empty set is a set that is waiting to be populated with elements.

Q: Can an empty set be equal to another set?

A: Yes, an empty set can be equal to another set if both sets have no elements. In other words, two empty sets are equal if they are both empty.

Q: Can an empty set be a subset of another set?

A: Yes, an empty set can be a subset of another set if the other set has no elements. In other words, an empty set is a subset of any set that has no elements.

Q: Can an empty set be a superset of another set?

A: No, an empty set cannot be a superset of another set. A superset is a set that contains all the elements of another set, and an empty set has no elements to contain.

Q: Can an empty set be equal to a set with elements?

A: No, an empty set cannot be equal to a set with elements. An empty set has no elements, whereas a set with elements has at least one element.

Q: Can an empty set be a subset of a set with elements?

A: No, an empty set cannot be a subset of a set with elements. A subset is a set that contains all the elements of another set, and an empty set has no elements to contain.

Q: Can an empty set be a superset of a set with elements?

A: No, an empty set cannot be a superset of a set with elements. A superset is a set that contains all the elements of another set, and an empty set has no elements to contain.

Q: What is the cardinality of an empty set?

A: The cardinality of an empty set is 0, which means that the empty set has no elements.

Q: Can an empty set be a union of sets?

A: Yes, an empty set can be a union of sets if all the sets in the union are empty.

Q: Can an empty set be an intersection of sets?

A: Yes, an empty set can be an intersection of sets if all the sets in the intersection are empty.

Q: Can an empty set be a difference of sets?

A: Yes, an empty set can be a difference of sets if the set being subtracted is empty.

Q: Can an empty set be a Cartesian product of sets?

A: Yes, an empty set can be a Cartesian product of sets if one or both of the sets in the product are empty.

Q: Can an empty set be a power set of a set?

A: Yes, an empty set can be a power set of a set if the set has no elements.

Q: Can an empty set be a set of sets?

A: Yes, an empty set can be a set of sets if the set contains no sets.

Q: Can an empty set be a set of functions?

A: Yes, an empty set can be a set of functions if the set contains no functions.

Q: Can an empty set be a set of relations?

A: Yes, an empty set can be a set of relations if the set contains no relations.

Q: Can an empty set be a set of ordered pairs?

A: Yes, an empty set can be a set of ordered pairs if the set contains no ordered pairs.

Q: Can an empty set be a set of sequences?

A: Yes, an empty set can be a set of sequences if the set contains no sequences.

Q: Can an empty set be a set of matrices?

A: Yes, an empty set can be a set of matrices if the set contains no matrices.

Q: Can an empty set be a set of vectors?

A: Yes, an empty set can be a set of vectors if the set contains no vectors.

Q: Can an empty set be a set of scalars?

A: Yes, an empty set can be a set of scalars if the set contains no scalars.

Q: Can an empty set be a set of complex numbers?

A: Yes, an empty set can be a set of complex numbers if the set contains no complex numbers.

Q: Can an empty set be a set of real numbers?

A: Yes, an empty set can be a set of real numbers if the set contains no real numbers.

Q: Can an empty set be a set of integers?

A: Yes, an empty set can be a set of integers if the set contains no integers.

Q: Can an empty set be a set of rational numbers?

A: Yes, an empty set can be a set of rational numbers if the set contains no rational numbers.

Q: Can an empty set be a set of algebraic numbers?

A: Yes, an empty set can be a set of algebraic numbers if the set contains no algebraic numbers.

Q: Can an empty set be a set of transcendental numbers?

A: Yes, an empty set can be a set of transcendental numbers if the set contains no transcendental numbers.

Q: Can an empty set be a set of irrational numbers?

A: Yes, an empty set can be a set of irrational numbers if the set contains no irrational numbers.

Q: Can an empty set be a set of non-negative numbers?

A: Yes, an empty set can be a set of non-negative numbers if the set contains no non-negative numbers.

Q: Can an empty set be a set of positive numbers?

A: Yes, an empty set can be a set of positive numbers if the set contains no positive numbers.

Q: Can an empty set be a set of negative numbers?

A: Yes, an empty set can be a set of negative numbers if the set contains no negative numbers.

Q: Can an empty set be a set of even numbers?

A: Yes, an empty set can be a set of even numbers if the set contains no even numbers.

Q: Can an empty set be a set of odd numbers?

A: Yes, an empty set can be a set of odd numbers if the set contains no odd numbers.

Q: Can an empty set be a set of prime numbers?

A: Yes, an empty set can be a set of prime numbers if the set contains no prime numbers.

Q: Can an empty set be a set of composite numbers?

A: Yes, an empty set can be a set of composite numbers if the set contains no composite numbers.

Q: Can an empty set be a set of perfect squares?

A: Yes, an empty set can be a set of perfect squares if the set contains no perfect squares.

Q: Can an empty set be a set of perfect cubes?

A: Yes, an empty set can be a set of perfect cubes if the set contains no perfect cubes.

Q: Can an empty set be a set of Fibonacci numbers?

A: Yes, an empty set can be a set of Fibonacci numbers if the set contains no Fibonacci numbers.

Q: Can an empty set be a set of Lucas numbers?

A: Yes, an empty set can be a set of Lucas numbers if the set contains no Lucas numbers.

Q: Can an empty set be a set of Mersenne numbers?

A: Yes, an empty set can be a set of Mersenne numbers if the set contains no Mersenne numbers.

Q: Can an empty set be a set of Fermat numbers?

A: Yes, an empty set can be a set of Fermat numbers if the set contains no Fermat numbers.

Q: Can an empty set be a set of Catalan numbers?

A: Yes, an empty set can be a set of Catalan numbers if the set contains no Catalan numbers.

Q: Can an empty set be a set of Bell numbers?

A: Yes, an empty set can be a set of Bell numbers if the set contains no Bell numbers.

Q: Can an empty set be a set of Stirling numbers?

A: Yes, an empty set can be a set of Stirling numbers if the set contains no Stirling numbers.

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