Union of Sets

A set is an organized collection of distinct elements, where every element is listed only once and that is represented as a separate element.

Example: The expression {0, 2, 4, 6, 8} can be used to represent a group of even numbers that are smaller than 10. There are five different elements in this set, and the order of elements does not matter. 
 

When two or more sets are joined together, the combination of all their elements without repetition is known as the union of sets. The symbol “∪” is used to indicate it. 

Example:
If A = {7, 8, 1} and B = {1, 0, 3}.

Then,
A ∪ B = {0, 1, 3, 7, 8}  

• How to find the union of sets? 

To better understand how to find the union of sets, let’s examine the following example. Since A = {c, d, e, f} and B = { x, d, y, z}, we have two sets, A and B. We must determine the components that make up the union of A and B.

According to the definition of the union of two sets, the resulting set includes all elements that are in set A, set B, or both.

• Set A = {c, d, e, f}
• Set B = {x, d, y, z}

Therefore, the union is {c, d, e, x, y, z}
Though d appears in both sets, it must be written only once because sets in a union do not include duplicate elements. 

• How to represent union of sets in Venn diagram

Venn diagrams are used to illustrate or clarify the connections between the specified set of operations. They use circles to symbolize each set.

Let us examine how to depict the union of two sets to find the union of sets using the Venn diagram. The two provided sets, H and M, are subsets of a universal set, which is needed to illustrate their relationship. This union of the sets H and M is shown in the Venn diagram.

The union of sets H and M (which is H ∪ M) is represented by the blue area in the Venn diagram above. This includes all elements that are in H, in M, or both sets.

The Venn diagram is frequently used to depict the union between multiple sets, as long as the sets are finite, even though the union operation between two sets has been used here.

Example:
K ∪ U = {2, 5, 6, 7, 8, 9}
K =  {2, 5, 6}
U =  {7, 8, 9}

If set K contains {2, 5, 6} and set U contains {7, 8, 9}, then the union K ∪ U would be 
{2, 5, 6, 7, 8, 9}. 

Take two sets, A and B, such that the following formula can be used to determine how many elements are in the union of A and B.
\(n(A ∪ B)=n(A)+n(B)−n(A ∩ B)\)

Here,
 

• A set’s correlation is defined as n(A ∪ B)= Total number of elements in A ∪ B.
   
• The value of a set A is defined as n(A)=Number of elements in A.
   
• The value of a set B is defined as n(B)=Number of elements in B.
   
• n(A ∩ B)=The number of elements shared by A and B are called the intersection of set A and B, or the intersection of B.

• Properties of Union of Sets

1. Commutative Law:
   When two or more sets are joined, the commutative law is followed; for example, if we have two sets A and B, then,
    A ∪B=B ∪ A
   
   Example:
   H = {b, c} and L= {c, f, a}
   H ∪ L = {b, c, f, a} and,
   L ∪ H = {c, f, a, b}
   
   Since the group of elements is the same in both unions, the commutative law is satisfied.
    A ∪ B=B ∪ A
    
2. Associative Law:
   The associative law governs the union operation, meaning that if we have three sets A, B, and C, then,
   (A ∪ B) ∪ C=A ∪ (B ∪ C)
   
   Example:
   H = {b, f} and L = {f, e, l} and M = {b, e, n}
   (H ∪ L) ∪ M = {b, f, e, l} ∪ {b, e, n} = {b, f, e, l, n}
   H ∪ (L ∪ M) = {b, f} ∪ {b, f, e, l, n}
   
   Hence, the associative law is proved.
   (A ∪ B) ∪ C=A ∪ (B ∪ C)
    
3. Identity Law:
   When any empty set is joined with any set A, the result is the set itself.
   A ∪ ∅ = A
   
   Example:
   H = {b, h, y} and ∅ = {}
   Then, H ∪ ∅ = {b, h, y} ∪ {} = {b, h, y}
   
   Hence, Identity law proved.
   A ∪ ∅ = A
    
4. Idempotent Law:
   Any set A that is joined to itself results in set A.
   A ∪ A = A
   
   Example: Assume that,
   H = {5, 4, 2, 1, 8}
   Then, H ∪ H = {1, 2, 4, 5, 8} ∪ {1, 2, 4, 5, 8}
   = {1, 2, 4, 5, 8} = H.
   
   Hence, the Idempotent law is proved.
   A ∪ A = A
    
5. Domination Law:
   The universal set itself is obtained by joining a universal set U with its subset A.
   A ∪ U = U
   
   Example: Assume that 
   A = {0, 2, 5, 6} and U = {0, 2, 4, 6, 7, 8, 9}.
   Then, A ∪ U = {0, 2, 5, 6} ∪ {0, 2, 4, 6, 7, 8, 9}  = {0, 2, 4, 6, 8, 9} = U
   
   Hence, proved.
   A ∪ U = U

Here are a few essential tips and tricks for students and parents:

1. You can use real-life situation to understand union of sets.
2. Learn the properties of sets.
3. Memorize the formula \(n(A ∪ B)=n(A)+n(B)−n(A ∩ B)\)
4. Express sets using Venn diagram for better visualization.
5. Practice all solved example by yourself.

Union of sets is used in real-life situations like combining survey data, analyzing students' participation, or merging customer lists. Here are a few real-world applications of the union of sets. 

 

1. Managing Hospital Patient Data
   If two lists in a hospital database contain patients with diabetes and hypertension, respectively, the union of these lists creates a complete list of all patients with any condition.
    
2. Combining social media contacts
   The union of sets makes it easier to combine a list of contacts or followers from various social media sites (such as Facebook, Instagram, Twitter, etc.) into one list without repeating anyone.
    
3. Categorization of library books
   Books in a library are arranged according to genres, such as science, fiction, and mystery. The union of sets makes it easier for readers or librarians to find all books in at least one of the categories.
    
4. Consolidation of shopping lists
   The union of sets helps to combine all the items into a single, comprehensive list without duplicating elements when two or more people create different shopping lists.
   
    
5. Students participating in clubs
   Every student who participates in one or both of these sports is a member of the union of these sets.