Venn Diagram

A Venn diagram is a visual representation showing how two or more sets of elements are related to each other. It helps us understand how different elements overlap and interact, making it simple for us to solve problems that involve things of different groups. Venn diagrams use circles to represent sets; based on whether they intersect or not, we can see which elements are shared or distinct among the sets. If the circles intersect, then it means there are common elements in two different sets. If they don’t intersect, then we can conclude that the sets have unique elements. 

To understand a Venn diagram, it is important to know key terms like universal set, subset, union, and more. In this section, we will learn all about them. 

Universal Set

A universal set has all the elements from the sets that are relevant at any given time. In a Venn diagram, the universal set is represented using a large rectangle, which is denoted by the letters E or U. To represent other sets, circles are drawn within the rectangle. All the sets are subsets of the universal set. 
 

The image above is a Venn diagram where, 
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 3, 5, 7, 9}, which is the set of odd numbers from 1 to 10. 
Here, the set A is the subset of U, so it is shown as a circle inside the rectangle. 

Subset

The elements of a subset also belong to another set. In the Venn diagram, a subset is represented by placing a circle inside another circle. If set A is a subset of set B, it can be written as A ⊆ B. 
 

A shown in the image, A is the subset of B, A ⊆ B.

Several standard symbols are used to represent the different relationships between sets in Venn diagrams. It is important to know the meaning of these symbols because they help us understand how the sets behave.

The Union Symbol: The union is represented by the symbol ∪. Union means joining or combining two or more sets. A ∪ B includes elements that are in set A, set B, or both. For instance, if A = {1, 2, 3} and B = {5, 8, 9}, then A ∪ B = {1, 2, 3, 5, 8, 9}

The Intersection Symbol: The intersection is represented by the symbol ∩. The intersection of two or more sets has the elements that are common to the sets. A ∩ B includes the elements that are common to both sets A and B. For example, P = {5, 6, 7, 8}and Q = {1, 2, 4, 5, 8}, then P∩Q = {5, 8}

Complement: If A represents a set, then the complement of A is written as A′ or Ac. The complement of a set includes elements that are present in the universal set but absent from the set. For example, if U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, then Ac = {1, 3, 5}.  

Venn diagrams can be used to represent different set operations, such as: 

• Union of Sets
   
• Intersection of Sets
   
• Complement of a Sets
   
• Difference of Sets

The union of two or more sets is a set consisting of all the elements in the sets. For example, let’s consider two sets, A and B. Now, when A and B are combined, they form a union that consists of elements that are present in both A and B. It is written as A ∪ B = {x | x ∈ A or x ∈ B}. In a Venn diagram, the union of two sets A and B is represented by two circles that overlap; each circle represents a set.
 

The intersection of sets consists of all elements that are common to both sets. It can be represented as A ∩ B = {x | x ∈ A and x ∈ B}. In a Venn diagram, the area where A and B intersect is shaded or colored, indicating that A and B have something in common.  

The complement of a set includes all the elements in the universal set, but not in the set. It is represented as A′ = {x | x ∉ A}, in a Venn diagram, A′ is the area inside the rectangle, excluding the circle representing set A.

In sets A and B, the difference of set (A - B) means that we want elements that are only in A. In the Venn diagram, A - B represents only that part of A that is not touching B.  

A basic Venn diagram consists of a rectangle, which represents the universal set. The subsets are shown using circles within the rectangle. Let us look at it step-by-step.

Step 1: Arrange the data into sets

First, based on the category, sort the data into different sets. 

Step 2: Draw the universal set
Next, we draw the rectangle to represent the universal set and label the rectangle. 

Step 3: Based on the number of categories, draw circles
Circles are drawn to represent sets. If the sets have common elements, then the circles overlap. 

Step 4: Labeling the circles
Label each circle with the name of the corresponding set.

As we learned how to draw a Venn diagram, let’s draw a Venn diagram. For example, draw a Venn diagram to represent the fruits and vegetables for the following: apples, bananas, oranges, tomatoes, broccoli, lettuce, mangoes, cucumbers, and carrots. 

Step 1: Here, we can sort the data into fruits, vegetables, and both fruits and vegetables. 
Set U: Apples, bananas, oranges, tomatoes, broccoli, lettuce, mangoes, cucumbers, carrots
Set A (fruits): Apples, bananas, oranges, mangoes, cucumbers, tomatoes
Set B (vegetables): Broccoli, lettuce, carrots, cucumbers, tomatoes
Both fruits and vegetables: Tomatoes and cucumbers

Step 2: Draw a rectangle to represent the universal set and label it

Step 3: Draw two circles in such a way that they overlap. One circle should be labeled as “fruits,” and another circle should be labeled as “vegetables.”

Step 4: Label the fruit names in the fruit circle and the vegetables in the vegetable circle. In the overlapping area, mention both the fruits and vegetables. 

There are different formulas that we can use to find more about any given set. Let’s take a look at them one by one: 

The union of two sets can be found by using the formula:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Here, n(A) and n(B) are the number of elements in A and B, respectively.
Similarly, n(A ∪ B) and n(A ∩ B) tell us the number of elements in A ∪ B and A ∩ B, respectively.

The formula to find the union of three sets is: 
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

The formula to find the complement of set A is:
n(Ac) = n(U) - n(A), where n(U) is the number of elements in the universal set.

The formula to find the difference of sets is:
n(A - B) = n(A) - n(A ∩ B)

Venn diagrams are used in different fields, such as the education sector, finance, logic, and statistics. Here are some applications of the Venn diagram:

• In business, we use Venn diagrams to compare and analyze market segments. It is helpful to understand the customers, products, services, etc.

• In education, Venn diagrams are used to compare and contrast two or more ideas. It is used to understand the relationship between different sets and their intersection or unions.

• In analysis and research, we use Venn diagrams to compare data sets and find relationships between them.