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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