As a domain expert in the field of mathematics, particularly in set theory, I can provide a comprehensive explanation of the concepts of union and intersection, which are fundamental in understanding the relationships between sets.
Union and intersection are two distinct operations that can be performed on sets. Let's delve into each one to understand their differences.
### Union
The union of two sets is a new set that is formed by combining all the elements from both sets, but without any duplicates. In other words, if an element is present in either set or in both, it will be included in the union. The union operation is denoted by the symbol \( \cup \), and it can be represented as \( A \cup B \) or \( \text{Union}[A, B] \). Here are some key points about the union:
- Inclusiveness: The union includes all elements that are in either set.
- Uniqueness: The union set does not contain any duplicate elements.
- Order: The order of elements in the union does not matter.
- Representation: The union can be visualized as a Venn diagram where all the areas that represent at least one of the sets are shaded.

### Intersection
The intersection of two sets is a new set that contains only the elements that are common to both sets. If an element is not present in both sets, it will not be included in the intersection. The intersection operation is denoted by the symbol \( \cap \), and it can be represented as \( A \cap B \) or \( \text{Intersection}[A, B] \). Here are some key points about the intersection:
- Commonality: The intersection includes only elements that are in both sets.
- Exclusivity: The intersection does not include elements that are unique to one set.
- Order: The order of elements in the intersection does not matter.
- Representation: The intersection can be visualized as a Venn diagram where only the overlapping area between the two sets is shaded.

### Differences Between Union and Intersection

1. Elements Included: Union includes all elements from both sets without duplication, while intersection includes only the elements that are present in both sets.

2. Symbol Representation: Union is denoted by \( \cup \), and intersection by \( \cap \).

3. Size of Resulting Set: The union of two sets is generally larger than or equal to the size of the intersection of the same sets.

4. Operational Significance: Union is used to find the totality of elements from both sets, whereas intersection is used to find the commonality between the sets.

5. Empty Set: The union of a set with the empty set is the set itself, while the intersection of a set with the empty set is the empty set.

### Examples
To illustrate the concepts further, let's consider two sets:
- Set A = {1, 2, 3}
- Set B = {2, 3, 4}

The union of sets A and B would be:
\[ A \cup B = \{1, 2, 3, 4\} \]

The intersection of sets A and B would be:
\[ A \cap B = \{2, 3\} \]

### Conclusion
Understanding the union and intersection of sets is crucial in various mathematical disciplines, including algebra, probability, and statistics. The union operation expands the scope by combining all elements, while the intersection operation narrows down to the common elements. Both operations are essential for analyzing the relationships and properties of sets.

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The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as A -- B or -- A or B --. The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as A -- B or -- A and B --.read more >>