Venn Diagrams are named after English Logician John Venn

The Venn Diagrams are named  after  English  Logician  John  Venn (1834-1923)  to present  pictorial  representation. The Universal Set, say  U or X is denoted by a region enclosed by a ractangle and one or more sets say, A, B, C are shown through circuits or closed curves within these rectangles. These circuits or closed curves intersect each other if there are any common elements amongst them, If there are no common elements then they are shown separately as disjionts.

Several set relations can be easily shown by these diagrams. These are useful to illustrate the set relations, such as the subset, set relations and the set-operations such as intersection, union, complemention, etc. by using regions in aplane to indicate sets. But Venn Diagrams (also known as Venn Euler Diagrams) can not be used to prove any statements regarding sets, just as geometric figures can not be used to prove geometric theorems. They are more aids for searching appropiate proofs.    

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Different Types Of Set

Sets may be of vatrious types. we give below a few of them...
Finite Set:
When the elements of a set can be counted by finite number of elements then the set is called a finit set. The following are the examples of finite sets:

In all above sets the elements can be counted by a finite number. It should br denoted that a set containing a very large number of elements is also a finite set. Thus, the set of all human beings in india, the set of all integers between -1 crore and +1 crore all finite sets.

Infinite Set:
If the elements of a set can not be counted in a finite number, the set is called an infinite set. The following are examples of infinite sets:

In all above sets, the process of counting the elements would be endless, hence these are infinite sets.

Singleton:
A Set containing only one element is called a singleton or a unit set. For example...

Empty, Null Or Void Set:
Any set which has no elements in it is called an empty set, or a null set or a void set. The symbol used to denote an empty is a Greek letter (read as phi), i.e., zero with a slish through it. Here the rule or the property discribing a given set is such that no element can be included in the set. The following are a few examples of empty sets:

Thus a we may say that laying down an impossible condition for the formation of a set produces an empty set.

Equal Set:
Two sets A and B are said to be equal if every element of A is also an element of B, and every element of B is also an element of A , i.e.,
We find that A=B=C, as each set contains the same elements namely a, c, h, m, r irrespective of their order. Hence the sets are equal. It may be noted that the order of elements or the repetition of elements does not matter in set theory.

Equivalent Sets:
If the elements of one set can be put into one to one correspondence with the elements of another set, then the two sets are called equivalent sets. For example..
Here A and C are equal sets while A and B are equivalent sets.

Subsets:
If every element of a set A is also an element of a set B then set A is called subset of set B. Symbolically we write this relationship as..

Proper Set:
Set A is called proper subset of superset B if each and every element of set A are the elements of the set B and at least one element of superset B is not an element of set A. Symbolically this is written a ACB and is read as 'A is a proper set of superset B'

Family Of Sets:
If all the elements of a set are sets themselves then it is called a 'set of sets' or better term is a 'family of sets'

Power Set:
From a set containing n elements 2n subsets can be formed. The set consisting of these 2n subsets is called a power set. In other words, if A be a given set then the family of sets each of whose number is a subset of the given set A called the power set A and is denoted as P(S), For example..

Universal Set:
When analysing some particular situation we are never required to go beyond some particular well-defined limits. This particular well-difined set may be called the universal set for the particular situation. Now onwords, we will assume that we are well aware of the particular universal set under consideration. The universal set will generally be denoted by the symbol 'U'

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Methods Of Describing A Set

The expression of sets has to be compact and clear otherwise the basic quality of the set being well-defined and distinactive is lost. Broadly there can be two approaches: (i) to the elements called the extension method, or (ii) toindicate the nature or characteristics and limits within which the elements lie. The latter method is rather unavoid able if the elements are too numerous, or not real butonly conceptual. These two approaches have been named variously as:

• Tabular, Roster Or Enumeration Method
• Selector, Property builder Or Rule Method

Tabular Method:
Under this method we enumerate or list all the elements of the set within brackes. However, there is no rigidity about these, use of even parentheses ( ) or brackers has been there in many books.

1. A set of Vowels: A = { a, e, i, o, u }
2. A set of odd natural numbers: N = { 1, 3, 5,.... }
3. A Set of Prime Ministers: P = { Nehru, Indiara Gandhi, Obama, Bosh }

Selector Method:
Under this method the elements are not listed but are indicated by description of their characteristics. We may state some characteristics which an object must possess in order to be an element in the set.
Here we choose the letter x to represent an arbitrary element of the set and write.

1. A= {x| x is a vowel in English alphabet}
2. B= {x| x is an old natural number}
3. C= {x| x is a Prime Minister of India}

The vertical line “ | ” after x to be read as ‘such that’. Sometimes we use ‘:’ to denote ‘such that’, e.g.,

• A= {x| x is a vowel in English alphabet}

It will be clear form the above that the toolbar method is particularly useful when the elements are few in number while the set-builder method is more suitable when the elements are numerous.

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Definition Of Set And Elements Of A Set

A Set:
A set is a collection of well-defined and well-distinguished objects. From a set it is possible to tell whether a given object belongs to a set or not. The following are some illustrations of a set:

• The possible outcomes in the toss of die
• The Integers from 1 to 100
• The vowels in English alphabets

The basic characteristics of a set are that it should be well-defined; its objects or elements should be well- distinguished for easy recognition by description.

Elements Of A Set:
The object that makes up a set are called the members or elements of the set. It is almost a convention to indicate sets by capital letter, like A, B, C or X, Y, Z while the elements in the set by smaller or lower case letters, viz.., a, b, c or x, y, z. Now, to indicate that a particular element or object “ belongs to a set “ or “ a member of the set ” we use the Greek symbol capital epsilon ∈.
For example if x is the member of a set A, We shall indicate it symbolically as:
x∈A,i.e.,x is a member Or an elements of the set A

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Theory Of Sets

Introduction:
The statements in the first chapter were concerned with individual objects. In Set we deal with a group of objects which can be defined in terms of their distinctive characteristic, magnitudes, etc. However, both the logical statements and sets belong to the same class. In the case of logical statements, we had three Boolean operators, like Conjunction, Disjunction and Negation. In set theory these are called as Intersection ∩ , Union ∪ and Complementation { } respectively.
Both they play an important role in modern Mathematics. The logical statements and Truth tables help in designing the circuits to perform Boolean Operations, the sets are of much wider application, especially they help in preparing the programme for feeding into the machine. In almost whole of the Business Mathematics the set theory is applied in one form or the other.

Structure:

• A Set
• Elements of a Set
• Method of Describing a Set
• Types of Set
• Venn Diagram
• Operations on Sets
• Intersection of Sets
• Union of Sets
• Complement of Sets
• De-Morgan’s Law
• Difference of two Sets
• Symmetric Difference
• Algebra of Sets
• Duality
• Partition of a Set
• Regrouping of the Sets
• Number of Elements In Finite Sets
• Ordered Pair
• Cartesian Product
• Set Relations
• Properties of Relations
• Binary Relations
• Functions Or Mappings
• Types Of Mappings

Objectives: After studying this chapter, You should be able to understand

• Set, Elements of a Set, Method of Describing a Set, Types of Set, Venn Diagram, Operations on Sets
• Algebra of Sets
• Cartesian Product
• Set Relations and its Relations
• Binary Relations, Functions and Mappings

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