6. Which One Of The Following Is A Null Set?(a) $\{0\}$(b) $\{\{\}\}$(c) $\{\}$(d) $\{x \mid X^2+1=0, X \in \mathbb{R}\}$

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A null set, also known as an empty set, is a set that contains no elements. It is denoted by the symbol {}\{\} or ∅\emptyset. In this article, we will explore the concept of a null set and determine which of the given options represents a null set.

What is a Null Set?

A null set is a set that has no elements. It is a set that is empty, meaning it does not contain any objects, numbers, or other sets. The null set is often denoted by the symbol {}\{\} or ∅\emptyset. It is a fundamental concept in mathematics, particularly in set theory.

Properties of a Null Set

A null set has several properties that distinguish it from other sets. Some of the key properties of a null set include:

  • It has no elements.
  • It is a subset of every set.
  • It is a proper subset of every set.
  • It has a cardinality of 0.

Analyzing the Options

Now that we have a clear understanding of what a null set is, let's analyze the options given in the problem.

Option (a): {0}\{0\}

This option represents a set that contains a single element, which is the number 0. Since this set contains an element, it is not a null set.

Option (b): {{}}\{\{\}\}

This option represents a set that contains a single element, which is another set. The inner set is a null set, but the outer set contains this null set as an element. Therefore, this set is not a null set.

Option (c): {}\{\}

This option represents a null set, as it contains no elements. This is the correct answer.

Option (d): {x∣x2+1=0,x∈R}\{x \mid x^2+1=0, x \in \mathbb{R}\}

This option represents a set that contains all real numbers x such that x2+1=0x^2+1=0. However, there is no real number that satisfies this equation, as the square of any real number is non-negative. Therefore, this set is empty and represents a null set.

Conclusion

In conclusion, the correct answer is option (c) {}\{\}, as it represents a null set. However, option (d) {x∣x2+1=0,x∈R}\{x \mid x^2+1=0, x \in \mathbb{R}\} also represents a null set, as it is an empty set. Therefore, both options (c) and (d) are correct.

Key Takeaways

  • A null set is a set that contains no elements.
  • A null set has several properties, including being a subset of every set and having a cardinality of 0.
  • Option (c) {}\{\} represents a null set.
  • Option (d) {x∣x2+1=0,x∈R}\{x \mid x^2+1=0, x \in \mathbb{R}\} also represents a null set.

Final Thoughts

A null set, also known as an empty set, is a set that contains no elements. It is a fundamental concept in mathematics, particularly in set theory. In this article, we will answer some frequently asked questions about null sets.

Q: What is a null set?

A: A null set is a set that contains no elements. It is a set that is empty, meaning it does not contain any objects, numbers, or other sets.

Q: How is a null set denoted?

A: A null set is often denoted by the symbol {}\{\} or ∅\emptyset.

Q: What are the properties of a null set?

A: A null set has several properties, including:

  • It has no elements.
  • It is a subset of every set.
  • It is a proper subset of every set.
  • It has a cardinality of 0.

Q: Is a null set the same as a set with a single element that is the empty set?

A: No, a null set and a set with a single element that is the empty set are not the same. A null set contains no elements, while a set with a single element that is the empty set contains one element, which is the empty set.

Q: Can a null set be a subset of another set?

A: Yes, a null set can be a subset of another set. In fact, a null set is a subset of every set.

Q: Can a null set be a proper subset of another set?

A: Yes, a null set can be a proper subset of another set. In fact, a null set is a proper subset of every set.

Q: What is the cardinality of a null set?

A: The cardinality of a null set is 0.

Q: Can a null set be equal to another set?

A: Yes, a null set can be equal to another set if and only if the other set is also a null set.

Q: Can a null set be a member of another set?

A: Yes, a null set can be a member of another set. In fact, a null set can be a member of any set.

Q: Can a null set be a union of other sets?

A: Yes, a null set can be a union of other sets. In fact, the union of any sets is always a set that contains all the elements of the individual sets, and if all the individual sets are null sets, then the union is also a null set.

Q: Can a null set be an intersection of other sets?

A: Yes, a null set can be an intersection of other sets. In fact, the intersection of any sets is always a set that contains all the elements that are common to the individual sets, and if all the individual sets are null sets, then the intersection is also a null set.

Q: Can a null set be a difference of other sets?

A: Yes, a null set can be a difference of other sets. In fact, the difference of any sets is always a set that contains all the elements that are in one set but not in the other set, and if one of the sets is a null set, then the difference is also a null set.

Q: Can a null set be a Cartesian product of other sets?

A: Yes, a null set can be a Cartesian product of other sets. In fact, the Cartesian product of any sets is always a set that contains all the ordered pairs of elements, one from each set, and if one of the sets is a null set, then the Cartesian product is also a null set.

Q: Can a null set be a power set of another set?

A: Yes, a null set can be a power set of another set. In fact, the power set of any set is always a set that contains all the subsets of the original set, and if the original set is a null set, then the power set is also a null set.

Conclusion

In conclusion, a null set is a fundamental concept in mathematics, particularly in set theory. It has several properties and applications, and it can be a subset, proper subset, member, union, intersection, difference, Cartesian product, or power set of other sets. By understanding the concept of a null set, we can better appreciate the beauty and complexity of mathematics.