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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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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