This article explains in detail two important basic concepts in set theory: intersection and union. We will define these two concepts separately, explain their differences and connections in concise and clear language and examples, and explore their wide range of uses in practical applications. By reading this article, you will be able to easily understand intersection and union, use them to solve related set problems, and improve your mathematical thinking skills.
An intersection is a new set composed of elements contained in two or more sets. Usually represented by the symbol "∩". For example, if there are sets A and B, their intersection is denoted as A ∩ B, which contains all elements that belong to both A and B.
Example
Assume that set A contains {1, 2, 3, 4}, set B contains {3, 4, 5, 6}, and their intersection A ∩ B contains {3, 4}, because these elements exist in both A and B middle.
A union is a new set composed of all the distinct elements from two or more sets. Usually represented by the symbol "∪". For example, if there are sets A and B, their union is denoted as A ∪ B, which contains all the different elements in A and B.
Example
Assume that set A contains {1, 2, 3, 4}, set B contains {3, 4, 5, 6}, and their union set A ∪ B contains {1, 2, 3, 4, 5, 6}, This is the combination of all the different elements in A and B.
The intersection contains the common elements of two or more sets, while the union contains all the different elements.
The result of intersection is a new set whose elements are all present in the original set, while the result of union is a new set whose elements are the combination of all the different elements in the original set.
Intersection is often used to solve problems with common attributes or conditions. For example, find elements in two collections that meet certain conditions.
Union is often used to merge data, such as merging elements from two lists or sets, to remove duplicates and get the complete set.
1. What is the basic difference between intersection and union?
The basic difference lies in the way they handle the relationships of elements in a collection. An intersection is a new set made up of the common elements from two or more sets, while a union is a new set made up of all the different elements in two or more sets.
2. How to express intersection and union?
Usually, the intersection is represented by the symbol "∩" and the union is represented by the symbol "∪". For example, the intersection of two sets A and B is represented as A ∩ B, and the union is represented as A ∪ B.
3. Can you give an example of practical application?
When processing database queries, intersection can be used to find records that meet multiple criteria, such as finding people who are between 30 and 40 years old and in a certain region. Union can be used to merge two data sets to obtain all data from different sources while removing duplicates.
4. Is it possible for a set to contain both intersection and union?
Yes, a set can contain both intersection and union. For example, if set A contains {1, 2, 3} and set B contains {2, 3, 4}, then A ∩ B is {2, 3} and A ∪ B is {1, 2, 3, 4} .
5. Are there any other set operations related to intersection and union?
Yes, set operations also include concepts such as complement set and difference set. The complement set refers to a set composed of elements in one set that do not belong to the other set. The difference set refers to the result of one set after removing the elements in the other set. These operations can be used to further manipulate the relationships between elements in a collection.
After studying this article, I believe you have a clear understanding of the concepts of intersection and union. Proficient in these two concepts will help you better understand and solve mathematical problems related to sets, and use them flexibly in practical applications.