Union of Sets

The union of sets is a foundational concept in set theory, widely used in mathematics, computer science, statistics, and real-world data analysis. Understanding the union operation helps you combine information, solve problems involving groups, and visualize relationships using Venn diagrams.

Union of Sets Definition & Meaning

The union of sets definition in maths states that the union of two or more sets is the set containing all elements from each set, without any duplicates. In simple terms, the union of sets meaning is to combine all unique elements from the involved sets into a single, comprehensive set.

• Formal Definition:
  For any sets A and B, the union is written as:
  A ∪ B = {x : x ∈ A or x ∈ B}

This means every element that appears in either set A, set B, or both, will be included in the union.

Union of Sets Symbol

The union of sets symbol is “∪” (a capital U-shaped symbol).

• A ∪ B is read as “A union B” and represents the set of all elements in A, in B, or in both.

Union of Sets Formula

The union of sets formula is used to calculate the total number of unique elements in the union of two sets:

n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)

Where:

• n(A∪B)n(A∪B) is the number of elements in the union
• n(A)n(A) and n(B)n(B) are the numbers of elements in sets A and B respectively
• n(A∩B)n(A∩B) is the number of elements common to both sets

For three sets, the formula is:

n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)

Union of Sets Venn Diagram

A union of sets Venn diagram visually represents the union by shading all areas belonging to either set. For two sets A and B, the combined area of both circles (including the overlap) is shaded to show all elements in A, B, or both.

Union of Sets Examples

Here are some union of sets examples to illustrate the concept:

• Example 1:
  If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then
  A ∪ B = {1, 2, 3, 4, 5, 6}
• Example 2:
  If X = {a, b, c}, Y = {b, d, e}, then
  X ∪ Y = {a, b, c, d, e}
• Example 3:
  If A = {red, blue}, B = {blue, green}, C = {green, yellow}, then
  A ∪ B ∪ C = {red, blue, green, yellow}

Union of Sets and Intersection of Sets

While the union of sets combines all unique elements, the intersection of sets includes only those elements that are present in all the involved sets.

• Union: A ∪ B = {x : x ∈ A or x ∈ B}
• Intersection: A ∩ B = {x : x ∈ A and x ∈ B}

The union is represented by the “∪” symbol, and the intersection by “∩”.

Properties of Union of Sets

• Commutative: A ∪ B = B ∪ A
• Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
• Identity: A ∪ ∅ = A
• Idempotent: A ∪ A = A
• Universal Set: A ∪ U = U

Real-Life Applications of Union of Sets

• Combining survey responses from different groups
• Merging customer lists in business analytics
• Unifying search results from multiple sources
• Database queries and data science