91Ó°ÊÓ

In set theory, the union of sets is a fundamental concept that helps us combine different sets together. The union of two sets, say \( A \) and \( B \), is denoted by \( A \cup B \). This operation results in a new set that contains all the elements that are in either set \( A \), set \( B \), or in both.
To visualize this, imagine listing all members of set \( A \) and then all members of set \( B \). The union \( A \cup B \) will include every single one of those elements, without any repetition.

• The union operation is inclusive; it gathers all possible members from the participating sets.
• It does not allow duplicate members—each element is unique within a set.

This property allows us to see how sets interact with one another, and is often used to prove relationships and formulate set-based equations. When working with union, always think of combining, collecting, and enlarging the group of elements.

An empty set is a special set in mathematics, represented by the symbol \( \emptyset \). This set contains absolutely no elements. One can think of it like an empty basket—it exists as a basket but holds nothing inside.

• The empty set is smartly used in mathematical proofs and problems, including operations like unions and intersections.
• Despite holding no elements, the empty set is important because it serves as the unique set that is a subset of every possible set.

In our problem, when an empty set is united with any other set \( A \), the result \( A \cup \emptyset \) is simply the original set \( A \).
No new elements are added by the empty set, because it contributes nothing to the union. Understanding the role of the empty set helps us respect how certain operations in set theory result in simplifications rather than complexities.

Set equality is a straightforward yet crucial concept in set theory. Two sets \( A \) and \( B \) are equal if they contain exactly the same elements. If every element of \( A \) is in \( B \) and every element of \( B \) is in \( A \), then we can declare \( A \) and \( B \) equal.
To prove set equality in practice, you often need to show two things:

• Every element in the first set is included in the second set.
• Every element in the second set is included in the first set.

In the specific case of the union with an empty set, \( A \cup \emptyset = A \), we confirm equality by proving that:1. All elements in \( A \) are in \( A \cup \emptyset \).2. There are no extra elements in \( A \cup \emptyset \) that are not in \( A \), thanks to the empty set providing no new entries.
By proving both these points, we establish that the union does not alter the original set, and thus, \( A \) remains unchanged when united with \( \emptyset \). Understanding set equality ensures that operations handling sets preserve or correctly change the sets involved when desired.