Venn Diagram in Set Theory

Introduction

Usually, the Venn diagrams represent a set. It is merely a pictorial relationship among either two or more sets where everything apart from the sets' elements is known as a universal set. These Venn diagrams were originated around 1880 by John Venn, a philosopher and English logician. They are used extensively to teach Set Theory Venn diagrams. A Venn diagram is also called a Primary diagram, Set diagram, or Logic diagram. In general, the universal set is represented by a rectangular region, disjoint sets by the disjoint circles, and intersecting sets by the intersecting circles. By universal sets, we mean the maximum possible number of points that can be included in any of the sets.

Representation of Sets using Venn Diagrams

• Each of the individual sets is mostly represented by a circle and enclosed within a quadrilateral (quadrilateral depicts the finiteness of the Venn diagram and the Universal set as well)

• Labelling is done for each of the sets with the name of the set to indicate the difference, and the respective constituting elements of each set are written within the circles

• Sets with no element in common are separately represented while those having some of the elements common within them are represented with overlapping

• The elements are written within the circle showing the set containing them, and the common elements are written in the parts of circles which are overlapped.

Let us look at some advanced venn diagram problems.

Case 1: When the Universal Sets and a Normal Set Have Been Given.

Let U is the universal set representing the all natural number’s sets and let X ⊆ U for X = 1,2,3,4,5

Then by Venn diagram, we can represent as,

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Case 2: When the Two Intersecting Subsets of U are Given.

For the representation of two intersecting subsets A and B of U, we usually draw two circles and the intersecting regions will be represented as:

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Case 3: When The Two Disjoint Sets are Being Given.

Normally, to draw the two disjoint sets, we draw two circles which will not intersect each other within a rectangle.

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Case 4: When B⊆A⊆U.

In this type of case, we draw the two concentric circles within a rectangular region, which can be represented by,

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Case 5: Complement of a Set

Let us consider a set as set A and it can be given as A = {1,2,3} i.e. A is the set of natural numbers.

Then the complement of A will be represented by,

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We present the subset value of one set by the venn diagram as below.

Let us take Y ⊆ X

If a ϵ Y, Y ⊆ X, then it also shows that a ϵ X.

Venn Diagram Example

Let us look at the example of advanced venn diagram problems and how to solve it using a venn diagram.

Problem 1

Suppose that, in a class, there are 50 students, where 10 take Guitar lessons, 20 take singing classes, 4 take both. Find out the number of students who do not take either Guitar or singing lessons.

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Solution

Let A denotes the number of students who take guitar lessons = 10

Let B denotes the number of students who take singing lessons = 20

Let C denotes the number of students who take both = 4

Now let us subtract the value of C from both A and B. Also, let us store the new values in D and E.

Therefore, by the given inputs, from the question,

D = 10 – 4 = 6

E = 20 – 4 = 16

Now, the logic represents that if we add the values, C, D, E and the unknown count, “X”, we should get a total of 50. Yes. That’s right.

So the final answer will be, X = 50 - C - D - E

X = (50) - (4) - (6) - (16)

X = 24

The Venn diagrams are specifically helpful in solving the word problems on number operations which involves counting. When it is drawn for a given problem, the rest should become a piece of cake.

Problem 2

In a college, there are 200 students, who are selected randomly. On those, 140 like tea, 120 like coffee and 80 like both the tea and coffee. So, find out:

• How many of the students like only tea?

• How many of the students like only coffee?

• How many of the students like neither tea nor coffee?

• How many of the students like only one of tea or coffee?

• How many of the students like at least one of the beverages?

Solution

The information given in the question can be represented using the following Venn diagram, where T = tea and C = coffee.

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Where,

X is the number of elements belong to only set A 

Y is the number of elements that belong to only set B

Z is the number of elements that belong to both set A, B, and set (AB)

W is the number of elements that belong to none of the sets of either A or B

From the above figure,

n(A) = x + z,

n (B) = y + z,

n (A ∩ B) = z,

n ( A ∪ B) = x+y+z

Therefore, the total number of elements = x + y + z + w

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So, by the representation given, we can define the values as,

• Students who like only tea is 60

• Students who like only coffee is 40

• Students who like neither tea nor coffee is 20

• Students who like only one tea or coffee is 60 + 40 = 100

• Finally, the students who like at least either one of tea or coffee = n (only Tea) + n (only coffee) + n (both Tea and coffee) = 60 + 40 + 80 = 180