Last updated on July 4th, 2025
Also known as a logic diagram or a set diagram, a Venn diagram is a tool that visually demonstrates the relationships between different groups of things or sets. Venn diagrams are used in various fields, including in mathematics, set theory, business, logic, and computer science. In this article, we will learn more about Venn diagrams.
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.
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:
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:
When drawing Venn diagrams, students tend to make mistakes. Here are some of the common mistakes and the ways to avoid them, and by learning these, students can master the Venn diagram.
Find the intersection of the sets M and N, where set M = {2, 4, 6, 8} and N = {6, 7, 8, 9}
M ∩ N = {6, 8}
Given,
M = {2, 4, 6, 8}
N = {6, 7, 8, 9}
M ∩ N is the set of elements common to both M and N; here, the common elements are 6 and 8.
So, M ∩ N = {6, 8}
If Set A = {1, 2, 3} and set B = {3, 4, 5}. Represent the union of A and B.
A ∪ B = {1, 2, 3, 4, 5}
To find the union of two sets, we first list all the elements of set A and B without repeating the elements. So A ∪ B = {1, 2, 3, 4, 5}
If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8} and set A = {2, 4, 6}. Find the complement of set A
The complement of set A = {1, 3, 5, 7, 8}
The complement of set A contains those elements present in the universal set and absent from set A.
Consider two sets: set A is the set of prime numbers between 1 and 20, and set B is the set of even numbers between 1 and 20. Draw a Venn diagram
NA
Set A = {2, 3, 5, 7, 11, 13, 17, 19}
Set B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
To draw the Venn diagram, we first draw a rectangle and then two overlapping circles. Here, 2 is a common element in both sets.
In a school, 12 players like bowling, 15 players like batting, and 5 players like both bowling and batting. Find the number of players like either bowling or batting.
The number of players who like either bowling or batting is the union
Here, the number of players who like bowling n(A) = 12
Number of players who like batting n(B)= 15
Number of players who like both bowling and batting n(A ∩ B) = 5
So, n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 12 + 15 - 5 = 22
So, the number of players who like either bowling or batting = 22
The number of players who like either bowling or batting includes those who like bowling, batting, and those who like both.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.