Union and Intersection of empty class of sets

In summary, the intersection of an empty class of sets is the whole space because there is nothing in the empty class that contradicts an element belonging to the intersection. However, the argument for the union is similar, where an element has to be in at least one of the sets. Since all the sets are empty, the union is also empty.
  • #1
AAQIB IQBAL
11
0
why intersection of empty class of sets is the whole space while their union is null set?
Book writes that an element will fail to be in the intersection if it fails to be in one of the sets of the class but since there is nothing in the empty class so there is nothing in the empty class that contradicts that an element does not belong to the intersection so in this way intersection is the whole space. ASSUME THIS IS TRUE. Then why don't they use the same arguments for the union? Then union should also have been the whole space...
 
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  • #2
The argument for the union is similar. To be in the union an element has to be in at least one of the sets. Since all the sets are empty, the union is empty.
 

1. What is the union of an empty class of sets?

The union of an empty class of sets is the set that contains all elements that are in at least one set in the class. Since there are no sets in the class, the resulting set is also empty.

2. What is the intersection of an empty class of sets?

The intersection of an empty class of sets is the set that contains all elements that are in every set in the class. Since there are no sets in the class, there are no elements that are common to all sets, and therefore the resulting set is also empty.

3. Can an empty class of sets have a union or intersection?

Yes, an empty class of sets can have a union and intersection. While the resulting sets will always be empty, the operations can still be performed on the class.

4. How does the union and intersection of an empty class of sets relate to the empty set?

The union and intersection of an empty class of sets are both equal to the empty set. This is because the empty set contains no elements, and therefore cannot be part of any set in the class.

5. What is the significance of the union and intersection of an empty class of sets?

The union and intersection of an empty class of sets are important concepts in set theory, as they demonstrate the properties of these operations when applied to an empty set. These operations also play a role in defining other set operations, such as the complement of a set.

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