Chapter I :  Set Language

   
Introduction: 

One of the most notable developments in mathematics is set theory. Set theory is used to compare the different branches of mathematics and make them as freely interchangeable as possible.  

Set theory was developed by the German mathematician George Cantor (1845-1918) as a very useful technique. It was devised to give mathematics a clearer look at the internal logic of their subject. By gaining an understanding of the ways sets operate, children can learn the elements common to arithmetic, algebra and geometry.  

Instead of dealing with numbers alone, Set theory deals with sets - well defined collection of things and instead of addition, subtraction and multiplication that are performed on numbers, the operations performed on sets are called unions, intersections and complementation.   

Sets are written in any one of following forms.  

1. Table form  

2. Roster form

3. Rule form 

4. Description form  

A set with a single element is called as a singleton set.  

A set with finite number of elements is called as FINITE SET. 

A set with infinite number of elements is called INFINITE SET. For example, the set of natural numbers is infinite. 

The number of distinct elements that a set contains is called the cardinal number of that set. 

The power set of a set is a set which contains all the subsets of that set. If a set contains  n number of elements, then its power set contains 2ⁿ elements. If n(A) = n then n(P(A)) = 2ⁿ.   

set which does not contain any elements is called the empty set, denoted by Φ. The empty set is a subset of every set.

The Venn diagram developed by John Venn is 1880 is often used as an aid in visualizing sets.   

The union of set A with set B (read “A union B” and written as 
(A U B) is represented by the region bounded by the two circles. The region where they overlap, labeled C represents the intersection of the two sets and is read “ A intersection B” and is written as A U B. It refers to what is common to both sets. The complement of A U B is D and it represents what is not in A U B. Finally, the union of A, B and D would be E, a set represented by the entire rectangle.  

FUNCTIONS: 

Let A and B be any two non empty sets. Let Fresh call denote some rule which associates with each element of A, a unique element of B. Then, we say  f is a function or a mapping from A to B ; denoted by f: A à B.

If an ‘x’ and a ‘y’ can be related through an equation or graph, they are called ‘variables’ : that is, one changes in value as the other changes in value. The two have what is known as a ‘functional relationship’. For example, the distance covered by a car is a ‘function’ of the speed with which it is driven, and the time taken to cover the distance.  

 Although seemingly of remote concern to anyone but the mathematician, the idea of variables and functions is of great importance. It has emerged as a concept fundamental to all higher mathematics.