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Sets And Venn Diagrams

Introduction

The branch of mathematical logic where we learn sets and their properties is known as set theory. The concept of set theory was initiated by German mathematician Georg Cantor (1845-1918) when he was working on “Problems on Trigonometric Series”. Set theory and its application are regarded as one of the fundamental concepts of mathematics. What are sets and how are they useful? Let us find out.

Definition

A set is a well-defined collection of objects. Here, by collection, we mean an aggregate of objects or things while aggregate itself means a class of things. Let a be an element of a set A. It is written as a belongs to A, which in symbolic form is written as a ∈ A.  If a is any other element but does not belong to the set A, it is written as a does not belong to A, which in symbolical form is written as a ∉ A. 

Let us understand it through an example. 

Suppose wish to create a collection of the vowels of the English alphabet. We know that the vowels are a, e, i, o and u. Hence, the collection of vowels in the English alphabet will be a, e, i, o and u. 

Some Standard Sets

In mathematics, we commonly refer to numbers as whole numbers, natural numbers etc. These numbers are standard sets of numbers based on their specific properties. Some of these standard sets are – 

Natural Numbers – The set of natural numbers consists of numbers starting from 1, 2 , 3, ………. and so on. It is represented by N.

Whole Numbers – The set of whole numbers consists of numbers starting from 0, 1, 2 , 3, ……… and so on. It is represented by W.

Integers – The set of integers consists of numbers such as …….-3, -2, -1, 0, 1, 2 , 3……. and so on. It is represented by Z.

Rational Numbers – The set of rational numbers consists of numbers that can be represented in the form of  $\frac{p}{q}$ where q ≠ 0. It is represented by Q.

Real Numbers – The set of integers consists of all real numbers such as …….-3, -2, -1, 0, 1, 2 , 3……. and so on. It is represented by R.

Description of a Set

A set is often described in the following two forms –

  1. Roster form
  2. Set builder form

Let us discuss these one by one.

Roster Form

In this form, a set is described by listing elements, separated by a comma and written within braces { }. 

Let us understand it through an example. 

Recall the set we described above for the vowels of the English alphabet. We obtained the elements of the set as a, e, i, o and u. let us name this set as A. This set in roster form will be represented as – 

A = { a, e, i, o and u } 

Similarly, the set of even natural numbers can be represented as  – 

A = { 2, 4, 6, 8, …… }

Set Builder Form

In the set builder form a set is described by a characteristic property P ( x ) of its elements x. In such a case the set is described by { x : P ( x ) holds } or { x | P ( x ) holds }, which is read as “ the set of all x such that P ( x ) holds. The symbol “ | “ is read as “ such that “.

Let us understand it through an example. 

Recall the set A = { a, e, i, o and u } which consisted of vowels of the English alphabet. This set in the set builder form will be written as  –

A = { x : x is a vowel in the English alphabet }

Similarly, the set of even natural numbers can be represented as  –

A = { x is a natural number and x = 2n for n N }

Types of Sets

Let us now learn about different types of sets depending upon the elements they contain. 

Empty set – A set is said to be an empty or null or void set if it has no element. Such a  set is denoted by ∅. In roster form, ∅ is denoted as { }.

For example, if we wish to create a set consisting of all natural numbers less than 1. We know that there does not exist any such natural number which is less than 1. Hence, this set will be represented as A  =  { x : x ∈ N and x < 1 } = ∅ or { }

Finite Set – A set is called a finite set if it is either void or it has a fixed number of elements. For example, the set of the first five natural numbers which will be given by A = { 1, 2, 3, 4, 5 } is a finite set as it has 5 fixed elements.

Infinite Set –  A set is called an infinite set if it does not have a  fixed number of elements. For example, the set of natural numbers which will be given by A = { 1, 2, 3, ………..  } is infinite as it has an infinite number of elements.

Equal Sets –  Two finite sets are said to be equal if they have the same elements. For example, the set A = { 1, 2 , 3, 4 } and the set B = { 1, 2 , 3, 4 } are equal.

Subsets –  Let A and B be two sets. If every element of A is an element of B, then A is called the subset of B. If A is a subset of B, we write it as A ⊆ B. Thus, if A ⊆ B, iff

a A ⇒ a ∈ B

For example,

{ 1 } ⊂ { 1, 2, 3 } but { 1, 4 } ⊄ { 1 , 2, 3 } 

Universal Set –  In any discussion in set theory, there always happens to be a set that contains all sets under consideration, i.e. it is a super set of each of the given sets. Such a set is called the universal set and is denoted by U. thus a set that contains all sets in a given context is called the universal set.

Venn Diagrams

Swiss mathematician, Euler, was the first to come up with a pictorial representation of sets. Later on British mathematician John Venn ( 1834 – 1883) brought this idea to practice. Named after these two mathematicians, we have Venn-Euler diagrams or simply Venn diagrams. Venn diagrams are the pictorial representation of the sets. In Venn diagrams, the universal set, U is represented by points within a rectangle. The subsets of the universal set are represented by points in circles within a rectangle. 

Let us understand it through an example. 

Subsets as Intersecting Circles

Let the universal set be the first 10 natural numbers. 

Hence, let U  = { 1, 2, 3, 4, 5, 6, , 8, 9, 10 }

Suppose we have two sets A and B where

A = { 1, 2, 3, 4, 5 } and B = { 1 , 3 , 5 , 7 , 9} 

We can see that the two sets A and B have some elements in common. Therefore, they will be represented as intersecting circles. The common elements shall be placed in the common region of the two circles. Also, the elements that belong to the universal set but are not a part of any subset shall be listed separately.

Subsets as Disjoint Circles

Again, let the universal set U  = { 1, 2, 3, 4, 5, 6, , 8, 9, 10 }

Suppose we have two sets A and B where

A = { 1, 2, 3, 4, 5 } and B = { 6, 7, 8, 9, 10} 

We can see that the two sets A and B have no element in common. Therefore, they will be represented as disjoint circles. The Venn diagram for this data will be represented as – 

Subsets as one circle within the other

Again, let the universal set U  = { 1, 2, 3, 4, 5, 6, , 8, 9, 10 }

Suppose we have two sets A and B where

A = { 1, 2, 3, 4, 5 } and B = { 3, 4 } 

We can see that all the elements of the set B are contained in the set A as well. This means that B ⊂ A. Since all the elements of the set are present in the set A, therefore, the set B will be shown inside the A with the common elements represented inside B. The other elements of A that are not present in B will be represented in the region that is not common to A and B. The Venn diagram for this data will be represented as – 

Operations on Sets

Let us now discuss various operations on sets using Venn diagrams – 

Union of Sets

Let A and B be two sets. The union of A and B is the set of all those elements which belong either to A or to b or to both A and B. The notation A B, read as A union B, is used to denote the union of two sets A and B. 

Thus A U B =  { x :  x ∈ A or x ∈ B }

Let us now learn how to represent the union of two sets through a Venn diagram. The shaded portion in yellow shown in the below Venn diagram represents the union of two sets A and B. 

Let us understand it through an example.

Suppose we have two sets A = { 1, 2, 3, } and B = { 1, 3, 5, 7 }

What will be A U B ? Let us find out.

A U B will be given by the union of all elements belonging either to A or to B. hence,

A U B = { 1 , 2, 3, 5, 7 }

Intersection of Sets

Let A and B be two sets. The intersection of A and B is the set of all those elements which belong to both A and B. The notation A ∩ B, read as A intersection B, is used to denote the intersection of two sets A and B. 

Thus A ∩ B =  { x :  x ∈ A and x ∈ B }

Let us now learn how to represent the intersection of two sets through a Venn diagram. The shaded portion in yellow shown in the below Venn diagram represents the intersection of two sets A and B. 

Let us understand it through an example.

Suppose we have two sets A = { 1, 2, 3, } and B = { 1, 3, 5, 7 }

What will be A ∩ B? Let us find out.

A ∩ B will be given by finding the common elements that belong both to A and B. Hence,

A ∩ B = { 3 }

Complement of a Set

Before we learn about the representation of complement of a set through the Venn diagram, let us learn what we mean by a complement of a set? Let U be the universal set and let A be a set such that A ⊂ U. then the complement of a with respect to U is denoted by A ‘ or A c or U – A and is defined as the set of all those elements of U which are not in A.

Let us understand it through an example.

Suppose we have universal set U  = { 1, 2, 3, 4, 5, 6, , 8, 9, 10 }

Let A be a subset of U such that A = { 1, 3, 5, 7, 9 }. Now the complement of A will be given by A ‘ = { 2, 4, 6, 8, 10 } which means all those elements that are present in the universals et but are not present in A.

Now, let us learn how to represent the complement of a set using Venn diagram. The shaded portion in yellow shown in the below Venn diagram represents the complement of a set A.

Laws of Algebra of Sets

Let us now learn about some fundamental laws of the algebra of sets.

  1. For any set A, the intersection of a set A with itself results in the same set, i.e. A U A = A
  2. For any set A, the union of a set A with empty set results in the same set, i.e. A U ∅ = A
  3. For any set A, the intersection of a set A with the universal set results in the same set, i.e. A ∩ U = A
  4. For any two sets A and B, the union of two sets A and B satisfies the commutative property, i.e. A U B = B U A
  5. For any two sets A and B, the intersection of two sets A and B satisfies the commutative property, i.e. A ∩ B = B ∩ A
  6. For any three sets A, B and C, the union of three sets A, B and C satisfies the associative property, i.e. ( A U B ) U C = ( A U ( B U C )
  7. For any three sets A, B and C, the intersection of three sets A, B and C satisfies the associative property, i.e. ( A ∩ B ) ∩ C = ( A  ∩ ( B ∩ C )
  8. For any three sets A, B and C, the union and intersection of three sets A, B and C satisfies the distributive property, i.e. ( A U ( B ∩ C ) = ( A U B )  ∩ ( A  ∪ C )

Key Facts and Summary

  1. A set is a well-defined collection of objects. 
  2. Let a be an element of a set A. It is written as a belongs to A, which in symbolic form is written as a ∈ A.
  3. In roster form, a set is described by listing elements, separated by a comma and written within braces { }.
  4. In set builder form a set is described by a characteristic property P ( x ) of its elements x.
  5. A set is said to be an empty or null or void set if it has no element.
  6. A set is called a finite set if it is either void or it has fixed number of elements.
  7. A set is called an infinite set if it does not have a fixed number of elements.
  8. The union of A and B is the set of all those elements which belong either to A or to b or to both A and B. The notation A U B, read as A union B, is used to denote the union of two sets A and B. 
  9. The intersection of A and B is the set of all those elements which belong to both A and B. The notation A ∩ B, read as A intersection B, is used to denote the intersection of two sets A and B.
  10. Let U be the universal set and let A be a set such that A ⊂ U. then the complement of a with respect to U is denoted by A ‘ or A c or U – A and is defined as the set of all those elements of U which are not in A.

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