The usage of Venn diagrams in mathematics, statistics, science, and engineering is widely known. While Venn diagrams ensure an easier representation of facts, they also aid the user in visualizing them. This ScienceStruck article helps you understand Venn diagrams with some examples, and also furnishes you with some free printable templates of the same.

###### Venn or Euler?

Venn and Euler diagrams look quite similar which makes it confusing to distinguish between the two. A Venn diagram shows all the possible combinations between sets even if there is no relation between them, whereas, a Euler diagram shows combinations only if they exist in the real world.

The name ‘Venn diagram’ is derived from its inventor John Venn. Widely used in mathematics, statistics, and engineering, it illustrates the relationship and the intersection between two or more sets. The intersection of sets defines the common elements between them. It is usually denoted by the ∩ symbol in set theory. While teaching mathematics, Venn diagrams are of great aid to teachers, as they help visualize the logical relations between sets. These diagrams are also used by professionals in making PowerPoint presentations to represent data or ideas. Let’s understand how to draw Venn diagrams with the help of a few examples and printable templates.

* Click on the blank templates to obtain a print.

Typically, a Venn diagram is drawn in a rectangle which denotes the universal set. Individual sets are denoted by circles that are placed in the bigger rectangle. The intersection of circles denotes the elements common to the two sets. You can draw a Venn diagram with any number of circles. This article gives examples and templates for the commonly used ones; namely, two, three and four-circle Venn diagrams.

**Examples**

**Using Venn Diagrams in Probability**

**Example**

100 students were asked which flavor of ice cream they preferred out of chocolate and vanilla.

65 students liked chocolate.

40 students liked vanilla.

10 students liked neither of the flavors.

Now if you were to randomly select a student from the group,

1. Find the probability of selecting a student who likes the chocolate flavor

2. Find the probability of selecting a student who likes both chocolate and vanilla flavors, given that you are picking from the set that likes either or both the flavors.

Before we solve these questions using Venn diagrams, let’s find the number of students who like both the flavors. Let *x* denote the number of students who like both the ice cream flavors. The total number of students is 100 and the number of students who like neither is 10. Thus, students who like either or both the flavors are 100 – 10 = 90.

Thus, **65+40- x = 90**

**∴**

*x*= 15Answer : Let ‘A’ represent the set of students who like chocolate flavor and ‘B’ represent the set of students who like both the flavors.

**1**

Out of the total students, probability of selecting a student who likes chocolate flavor is given as follows.

P (A) = 65/100 ≈ 0.65

**2**

It is already mentioned that the student picked is from the set that likes either or both the flavors. Thus, 10 students who like neither of the flavors need not be considered.

The probability of selecting a student who likes both the flavors, from the set of students who either like one flavor or both is given as follows.

P (B) = 15/90 ≈ 0.17

**Using Venn Diagrams in Logic**

Syllogism in ‘Categorical Logic’ makes an interesting use of Venn diagrams. They have three categories or propositions, which consist of two premises and one conclusion. The conclusion is deduced from the two premises given. Venn diagrams are popularly used to test the validity of these syllogisms. With the help of three overlapping circles, the conclusion is tested by diagramming the premises on these representative circles.

**Example**

Major premise: All M are P.

Minor premise: All S are M.

Conclusion: Therefore, All S are P.

The above-mentioned example has been denoted by the mnemonic B**a**rb**a**r**a** (AAA-1). All the aforesaid premises are ‘Universal Affirmatives’.

All snakes are reptiles.

All cobras are snakes.

Therefore, all cobras are reptiles.

Let ‘S’ represent all ‘Cobras’, ‘M’ represent ‘Snakes’ and ‘P’ represent all ‘Reptiles’. To test whether the syllogism’s conclusion is valid, let’s take help of a Venn diagram.

First, to represent, ‘All snakes are reptiles’, we shade out, that portion of circle M which is not in circle P. This indicates that all of circle M is in Circle P.

Secondly, to represent the minor premise, ‘All cobras are snakes’, we shade out that portion of Circle S which is not in Circle M.

To make it a valid syllogism, the conclusion should implicate into a result which is commonly said by the premises. Now, if we see the conclusion ‘ All cobras are reptiles’ – all of Circle S should be in Circle P, and while diagramming the two premises, that portion of Circle S which was not in Circle P has been shaded out automatically. Thus, the Venn diagram proves our syllogism to be valid.

**Using Venn Diagrams in Set Theory**

Set theory makes wide use of Venn diagrams. It is easy to understand the union and intersection of sets with the help of Venn diagrams.

**Example**

Set A contains multiples of 2

Set B contains multiples of 4

Set C contains multiples of 6

Set D contains multiples of 8

Set A = {2,4,6,8,10,12,14,16}

Set B = {4,8,12,16,20,24,28,32}

Set C = {6,12,18,24,30,36,42,48}

Set D = {8,16,24,32,40,48,56,64}

The intersection of all the sets, can be easily displayed in the common space of all the four circles.**A ∩ B ∩ C ∩ D = {24}**

Venn diagrams make understanding logic, math, and probability easier and more fun too. You can use the printable templates given here, to solve math problems using Venn diagrams. On a lighter side, you could use them to present ideas. For instance, success can be represented as an intersection set of passion, talent, and market demand. Or good business leaders can be represented as an intersection of sets representing people who dream big, and people who can take risks. To represent something funny, like ‘all women love shopping’, you can have a circle denoting ‘all women’ drawn inside a circle denoting ‘people who love shopping’. Be it school science or real-world scenarios, Venn diagrams are of great help in representing data sets and explaining the relations between them.