# How to use a Venn diagram?

Here is how you can use the Venn diagram in several contexts to make your teamwork more efficient and collaborative.

## What is a Venn diagram?

Some concepts, which can be extremely complex to grasp when merely explained, benefit from being laid out in the form of a graphical representation, or diagram. Venn diagrams are specialized in representing the relationships between data sets. They show the intersections between different sets of ideas, elements or concepts in a visual, simple and most importantly, snackable way.

## Why "Venn diagrams"?

While there is evidence that diagrams were used by philosophers (namely Ramon Llull of Majorca) in the 1200s, the official invention of the Venn diagram dates back to 1880. That year, the British mathematician and logician John Venn published his paper “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings”. Fast-forward to 1918, when the philosopher and logician Clarence Irving Lewis referred to Venn’s work, coining the term “Venn diagram”. This is when its use became more generalized.

Venn diagrams range from probability mapping, statistics, linguistics, set theory, computer science, and business to the natural sciences. From the elementary to the professional level, these representations are used to find specific patterns or behaviors in the intersections of the different components of the diagram and to derive new knowledge from it.

In a Venn diagram, each set of information is usually represented by a circle and, in some cases, by an outer rectangle indicating a universal set. When the circles are placed, they intertwine with each other, forming subsets that outline the commonalities between two independent concepts. We can then say that they are composed of three general elements:

• Intersections
• Unions
• Complementary sets

All the possible relations between the various elements of the set are clearly represented in how the circles overlap. This visual depiction makes it possible to understand at a glance what would otherwise demand multiple convoluted sentences to explain.

To put it simply, a Venn diagram very conveniently illustrates such notions as “all X are Y, but not all Y are X.

One could use the example that “all birds lay eggs but not all animals that lay eggs are birds”. A Venn diagram could show various groups of animals by dedicating a section (or circle) to birds, one to reptiles, one to fish, one to insects, and so on. Within these circles, several characteristics of the animal groups would be categorized. Where these characteristics intersect would clearly show what the different groups have in common. In our example, laying eggs would stand out as a part of the diagram that encompasses all four types of animals. Flying, on the other hand, would only apply to a subgroup of birds and a subgroup of insects, but not to the reptiles or the fish (generally speaking, of course). Having scales would apply to some of the reptiles and fish, but not to the other groups, etc.

## Universal sets

Interactions between sets are defined by the elements belonging to two or more sets, and are placed according to the number of sets you have. You can start from 1 and multiply by the number of sets you have chosen. It should be noted that all parts and elements must appear in the diagram. Therefore, if some elements do not belong to any intersection, they should be placed in the free part of the circle. The reason for this is to be able to observe the singularities of the unlinked elements and to perform a more thorough analysis.

## The different types of Venn diagrams according to their relationships

We have already pointed out that a Venn diagram is useful for seeing the relationships of overlapping sets, but there are other ways to use this type of diagram. In all cases, we can see the same elements, but it is in these overlaps that we can see differences.

When a diagram involves a greater number of sets, some loss of symmetry is inevitable. Venn was eager to find "symmetrical figures elegant in themselves " that represented a larger number of sets, so he designed a diagram consisting of four sets using ellipses (see below). He also gave a construction for Venn diagrams for any number of sets, where each successive curve that bounds a set intertwines with the preceding curves, starting with the three-circle diagram.

### Set overlays

This is the most common method for considering the relationships between different intersections. All elements of an intersection belong to two, three or more sets.

### Inclusion

All elements of a set also belong to another set, i.e. they form a subset. In this case, a small set belongs to or is contained within a larger set, so the intersection represents the subset itself. This type of diagram refers directly to an Euler diagram.

### Disjunction

Here, we talk about sets not having any elements in common. Even if the elements are plotted and intersections exist, no elements with a relationship are found, so these intersections are empty. This is very useful for disproving assumptions or correcting an analysis.

## Uses of the Venn diagram

Widely used in business, the Venn diagram is an excellent way to show the relationships between several groups of things or concepts. As it is conceived following basic geometrical figures, this diagram can be made even easier with an online tool, allowing it to be shared or integrated into a presentation, for example.

The Venn diagram has many advantages and, for this reason, it has been used for many years:

• It visually organizes information from basic to advanced, allowing it to meet a variety of needs while maintaining a simple presentation.
• It allows you to compare two or more options in order to understand what they have in common and what makes them different.
• It allows you to solve complex problems, especially in the field of mathematics.
• It allows you to find correlations and predict the probability of certain events.
• It allows us to understand the logic behind a formulation.

### Presentations

This is probably one of the most common uses of the diagram, as it is a great way to visually present some research, a summary, or a task already completed. You can make your diagram online and print it out afterward for greater impact.

### Research

We often have to research complicated topics, and sorting the information becomes a tedious task. While there are methods such as mind mapping, brainstorming and many other strategies, the Venn diagram is ideal for separating and analyzing the relationships between sets of information.

### Strategic analysis

By separating and connecting elements, comparisons can be made immediately. These comparisons ensure that the strategy is composed of elements that have a beneficial relationship with the final objective. Think of analyzing the relationship between soccer players, sales, stores, products, etc.

Discover more tools to conduct strategic analysis.

### Education

Many teachers use this diagram to highlight the differences and similarities between sets of topics seen in class. They are used in subjects such as mathematics, physics etc. And outside the classroom, they are also an excellent study tool.

### Natural Sciences

The analysis of relationships between sets is also widely used in the study of chemical phenomena, processes, mechanisms or biological elements for their study, teaching and application.

### Computer Science

Data sets are a very important part of computer science, as they represent the main values in the development of software, applications, code, etc. With this type of diagrams, experts can perform an analysis or group these data points according to what they want to analyze or develop.

Now you know what the Venn diagram is for and all the advantages it has. These are just a few examples. You can use any tool you want to make your diagram, whether it is a digital application, an online site, etc. How you want to use your Venn diagram depends on your needs and the objectives of your project.

## What is the relationship between digital Venn diagrams and web-based business?

Laying out user-related data in the form of a Venn diagram can reveal some very significant correlations that you can use to optimize your business strategy. For example, you may find that the number of users accessing your site from a desktop device far exceeds that of mobile users. And yet, once in the form of a visual representation, the figures may prove that the proportion of mobile users who meet your conversion goals is comparatively much higher. From that, you could deduce that it would be beneficial to focus your marketing efforts on reaching smartphone users, since there is a higher probability that they will be interested in your products or services.

### Probabilities

In probability, a Venn diagram is a figure with one or more circles inside a rectangle that describes the logical relationships between events. The rectangle in a Venn diagram represents the sample space or the universal set, that is, the set of all possible outcomes. A circle inside the rectangle represents an event, that is, a subset of the sample space.

We can also use a Venn diagram to calculate conditional probabilities.

We can also use a Venn diagram to describe the relationships between three events. In the two special situations listed below, we have a special representation of a three-event Venn diagram.

1. W. F. Edwards (en) has constructed a series of Venn diagrams for a larger number of sets, by segmenting the surface of a sphere. For example, three sets can be easily represented by taking three hemispheres of a sphere at right angles (x = 0, y = 0 and z = 0). A fourth set can be added to the representation by taking a curve similar to the seam of a tennis ball, and so on. These patterns were conceived in the design of a stained glass window in memory of Venn.
2. Edwards' Venn diagrams are topologically equivalent to the diagrams developed by Branko Grünbaum. They are also 2-dimensional representations of the hypercube.