Visual representation often helps in understanding a problem and improving mathematical reasoning. Venn diagrams in math are the visual representation of sets and their operations. Venn diagrams are also called logic diagrams or set diagrams and are widely used in set theory, logic, mathematics, businesses, teaching, computer science, and statistics.
Let’s understand what is a Venn Diagram in math and the symbols used to represent it, along with its properties.
What is a Venn Diagram?
Venn diagrams introduced by John Venn (1834-1883) are the diagrams that are used to represent the sets, the relations between the sets, and the operations performed on them, in a pictorial manner. Venn diagram uses circles (overlapping, intersecting, and non-intersecting), to denote the relationship between sets. Often, they serve to graphically organize things, highlighting how the items are similar and different.
A Venn diagram can be represented by any closed figure, such as a circle or any other polygon (like a square, rectangle, hexagon, etc.). But usually, we use circles to represent each set and a rectangle to represent a universal set in a Venn diagram.
In the above figure, a rectangle represents a universal set $\text{U}$ and a circle represents a set $\text{A}$.
Venn Diagram Examples
Example 1: Represent a set of natural numbers from $1$ to $10$ where the universal set is a set of natural numbers.
In the above figure set $\text{A} = \{x : x \text{ is a natural number and } x \le 1 \le 10 \}$ and $\text{N}$ is a set of natural numbers and is a universal set.
Example 2: Represent a set of even numbers and a set of odd numbers where the universal set is a set of natural numbers.
In the above figure, $\text{N}$ is a set of natural numbers and is a universal set, and set $\text{E}$ represents a set of even numbers and set $\text{O}$ represents a set of odd numbers.
Terms Related to Venn Diagrams
There are certain terms associated with Venn diagrams. Let’s understand these terms.
Universal Set in Venn Diagram
Whenever we use a set, it is easier to first consider a larger set called a universal set that contains all of the elements in all of the sets that are being considered. The following points are taken into consideration whenever we draw a Venn diagram.
- A large rectangle is used to represent the universal set and it is usually denoted by the symbol $\text{U}$.
- All the other sets are represented by circles or closed figures within this larger rectangle.
- Every set is the subset of the universal set $\text{U}$.
In the above figure,
- $\text{U}$ is the universal set with all the numbers $1$ to $10$, enclosed within the rectangle.
- $\text{A}$ is the set of odd numbers $1$ to $10$, which is the subset of the universal set $\text{U}$ and it is placed inside the rectangle.
- All the numbers between $1$ to $10$, that are not odd(i.e., even), will be placed outside the circle and within the rectangle.
Subset in Venn Diagram
Venn diagrams are used to show subsets. A subset is a set that is contained within another set. Let us consider two sets $\text{A}$ and $\text{B}$. Further, let $\text{A}$ be a subset of $\text{B}$. In the figure below, you can see that circle $\text{A}$ is contained entirely within circle $\text{A}$. Also, all the elements of $\text{A}$ are elements of set $\text{A}$.
The relationship of a subset is symbolically represented as $\text{A} \subset \text{B}$. It is read as $\text{A}$ is a subset of $\text{B}$.
In the above figure,
- $\text{R}$ = set of real numbers. (It’s a universal set)
- $\text{N}$ = set of natural numbers.
- $\text{I}$ = set of integers.
- Here $\text{N} \subset \text{I}$, because all natural numbers are integers(Natural numbers are positive integers).
Note: Every set is a subset of itself, i.e. $\text{A} \subset \text{A}$.
Venn Diagram and Sets Operations
With sets, we can perform certain set operations. These operations are
- Union of Set
- Intersection of set
- Complement of set
- Difference of set
Union of Sets Venn Diagram
The union of two sets $\text{A}$ and $\text{B}$ is given by $\text{A} \cup \text{B} = \{x : x \in \text{A or } x \in \text{B} \}$. This operation on the elements of set $\text{A}$ and $\text{B}$ can be represented using a Venn diagram with two circles. The total region of both the circles combined denotes the union of sets $\text{A}$ and $\text{B}$.
Intersection of Sets Venn Diagram
The intersection of sets, $\text{A}$ and $\text{B}$ is given by $\text{A} \cap \text{B} = \{x : x \in \text{A} \text{ and } x \in \text{B} \}$. This operation on set $\text{A}$ and $\text{B}$ can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denote the intersection of set $\text{A}$ and set $\text{B}$.
Complement of Set Venn Diagram
The complement of any set $\text{A}$ can be given as $\text{A}^{‘}$. This represents elements that are not present in set $\text{A}$ and can be represented using a Venn diagram with a circle. The region covered in the universal set, excluding the region covered by set $\text{A}$, gives the complement of $\text{A}$.
Difference of Set Venn Diagram
The difference of sets can be given as $\text{A} – \text{B}$. It is also referred to as a ‘relative complement’. This operation on sets can be represented using a Venn diagram with two circles. The region covered by set $\text{A}$, excluding the region that is common to set $\text{B}$, gives the difference of sets $\text{A}$ and $\text{B}$, i.e., $\text{A} – \text{B}$.
Examples of Venn of Set Operations
Example 1: In a class of $40$ students, $20$ have chosen Mathematics, $15$ have chosen mathematics but not biology. If every student has chosen either mathematics or biology or both, find the number of students who chose both mathematics and biology and the number of students who chose biology but not mathematics.
Let, $\text{M} be the set of students who chose mathematics.
$\text{B} be the set of students who chose biology.
Therefore, $n \left( \text{M} \cup \text{B} \right) = 40$ and $n \left( \text{M} \right) = 20$
$n (\text{B}) = n \left( \text{M} \cup \text{B} \right) – n(\text{M})$
$=> n(\text{B}) = 40 – 20 = 20$
$n(\text{M} – \text{B}) = 15$
$n(\text{M}) = n(\text{M} – \text{B}) + n \left( \text{M} \cap \text{B} \right)$
$=>20 = 15 + n \left( \text{M} \cap \text{B} \right)$
$=> n \left( \text{M} \cap \text{B} \right) = 20 – 15 = 5$
Therefore, the number of students who chose both mathematics and biology is $5$.
$n(\text{B} – \text{M}) = n(\text{B}) – n \left( \text{M} \cap \text{B} \right)$
$=> n(\text{B} – \text{M}) = 20 – 5 = 15$
Therefore, the number of students who chose biology but not mathematics is $15$.
Example 2: In a class, there are $15$ students who like chocolate. $13$ students like vanilla. $10$ students like neither. If there are $35$ people in the class, how many students like chocolate and vanilla?
Let, $\text{C} be the set of students who like chocolate.
And, let $\text{V} be the set of students who like vanilla.
Therefore, $n(\text{C}) = 15$, and $n(\text{V}) = 13$
Now, let’s draw the Venn diagram showing the intersection of chocolate and vanilla. The outside of the Venn diagram is $10$, and the total of the entire diagram must equal $35$.
Therefore the two circles of the Venn diagram including just chocolate, just vanilla, and the intersection must equal $25$, with the just chocolate plus intersection side equalling $15$ and the just vanilla plus intersection side equalling $13$.
We also know that $\left( \text{A} \cup \text{B} \right) = \text{A} + \text{B} – \left( \text{A} \cap \text{B} \right)$.
We have found that $\left( \text{A} \cup \text{B} \right) = 25$ and we are trying to find $\left( \text{A} \cap \text{B} \right)$.
Therefore, put in the values of $n(\text{A})$ and $n(\text{B})$.
$25 = 15 + 13 – \left( \text{A} \cap \text{B} \right) = 28 – \left( \text{A} \cap \text{B} \right)$
$=> – \left( \text{A} \cap \text{B} \right) = –3$
$=> \left( \text{A} \cap \text{B} \right) = 3$
Therefore, the number of students who like chocolate and vanilla is $3$.
Venn Diagram Purpose and Benefits
The following are the main benefits of using Venn diagrams.
- To visually organize information to see the relationship between sets of items, such as commonalities and differences. Students and professionals can use them to think through the logic behind a concept and to depict the relationships for visual communication. This purpose can range from elementary to highly advanced.
- To compare two or more choices and clearly see what they have in common versus what might distinguish them. This might be done for selecting an important product or service to buy.
- To solve complex mathematical problems. Assuming you’re a mathematician, of course.
- To compare data sets, find correlations, and predict probabilities of certain occurrences.
- To reason through the logic behind statements or equations, such as the Boolean logic behind a word search involving “or” and “and” statements and how they’re grouped.
Applications of Venn Diagram
Venn diagrams are widely used in many areas. Some of the most important uses of Venn diagrams are
- Math: Venn diagrams are commonly used in school to teach basic math concepts such as sets, unions, and intersections. They’re also used in advanced mathematics to solve complex problems and have been written about extensively in scholarly journals. Set theory is an entire branch of mathematics.
- Statistics and Probability: Statistics experts use Venn diagrams to predict the likelihood of certain occurrences. This ties in with the field of predictive analytics. Different data sets can be compared to find degrees of commonality and differences.
- Logic: Venn diagrams are used to determine the validity of particular arguments and conclusions. In deductive reasoning, if the premises are true and the argument form is correct, then the conclusion must be true. For example, if all dogs are animals, and our pet Mojo is a dog, then Mojo has to be an animal. If we assign variables, then let’s say dogs are C, animals are A, and Mojo is B. In argument form, we say: All C is A. B is C. Therefore B is A. A related diagram in logic is called a Truth Table, which places the variables into columns to determine what is logically valid. Another related diagram is called the Randolph diagram, or R-Diagram, after mathematician John F. Randolph. It uses lines to define sets.
- Linguistics: Venn diagrams have been used to study the commonalities and differences among languages.
- Teaching Reading Comprehension: Teachers use Venn diagrams to improve their students’ reading comprehension. Students can draw diagrams to compare and contrast ideas they are reading about.
- Computer Science: Programmers use Venn diagrams to visualize computer languages and hierarchies.
- Business: Venn diagrams are used to compare and contrast products, services, processes, or pretty much anything that can be depicted in sets. And they’re an effective communication tool to illustrate that comparison.
Key Takeaways
| Set: A collection of things. Given the versatility of Venn diagrams, things can really be anything. The things may be called items, objects, members, or similar terms. |  |
| Union: All items in the sets. |  |
| Intersection: The items that overlap in the sets. Sometimes called a subset. |  |
| Complement: The elements that are not present in the set. |  |
| Symmetric Difference: Everything but the intersection. |  |
Practice Problems
- If $\text{A}$ and $\text{B}$ are two sets such that number of elements in $\text{A}$ is $32$, the number of elements in $\text{B}$ is $28$ and the number of elements in both $\text{A}$ and $\text{B}$ is $10$, find:
- $n \left( \text{A} \cup \text{B} \right)$
- $n \left( \text{A} – \text{B} \right)$
- $n \left( \text{B} – \text{A} \right)$
- According to the survey made among $250$ students, $170$ students like cold drinks, $140$ students like milkshakes, and $100$ like both. How many students like at least one of the drinks? How many students do not like any?
- In a group of $500$ people, $350$ people can speak English, and $400$ people can speak Hindi. Find how many people can speak both languages.
FAQs
How do you explain a Venn diagram?
Venn diagram uses circles (overlapping, intersecting, and non-intersecting), to denote the relationship between sets. Often, they serve to graphically organize things, highlighting how the items are similar and different.
A Venn diagram can be represented by any closed figure, such as a circle or any other polygon (like a square, rectangle, hexagon, etc.). But usually, we use circles to represent each set and a rectangle to represent a universal set in a Venn diagram.
Is a Venn diagram always 3 circles?
No, a Venn diagram can consist of multiple intersections and circle sets. The number of circles represents the number of sets in a problem.
What is $\text{ A} \cap \text{B }$ Venn diagram?
$\text{A} \cap \text{B}$ in a Venn diagram represents intersection of two sets. The intersection of sets, $\text{A}$ and $\text{B}$ is given by $\text{A} \cap \text{B} = \{x : x \in \text{A} \text{ and } x \in \text{B} \}$. This operation on set $\text{A}$ and $\text{B}$ can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denote the intersection of set $\text{A}$ and set $\text{B}$.
What is $\text{ A} \cup \text{B }$ Venn diagram?
$\text{A} \cup \text{B}$ in a Venn diagram represents union of two sets. The intersection of sets, $\text{A}$ and $\text{B}$ is given by $\text{A} \cap \text{B} = \{x : x \in \text{A} \text{ and } x \in \text{B} \}$. This operation on set $\text{A}$ and $\text{B}$ can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denote the intersection of set $\text{A}$ and set $\text{B}$.
What is a $\text{C}$ in sets?
The letter $\text{C}$ represents the complement of a set. It is also denoted by a symbol $’$. If $\text{A}$ is a set, then the complement of set $\text{A}$ will contain all the elements in the given universal set ($\text{U}$), that are not in set $\text{A}$. It is usually denoted by $\text{A}^{‘}$ or $\text{A}^{c}$.
Conclusion
A Venn diagram is a pictorial representation of set(s) and their operations. Venn diagram uses circles (overlapping, intersecting, and non-intersecting), to denote the relationship between sets. They are widely used in set theory, logic, mathematics, businesses, teaching, computer science, and statistics.
