Sets

 

The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”. 
Sets and their Representations
 A set is a well-defined collection of objects. 
The following points may be noted :
(i) Objects, elements and members of a set are synonymous terms.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.
If a is an element of a set A, we say that “ a belongs to A” the Greek symbol ∈
(epsilon) is used to denote the phrase ‘ belongs to’. Thus, we write a ∈ A. If ‘ b’ is not an element of a set A, we write b ∉ A and read “b does not belong to A”.
Thus, in the set V of vowels in the English alphabet, a ∈ V but b ∉ V. In the set P of prime factors of 30, 3 ∈ P but 15 ∉ P.
There are two methods of representing a set :
(i) Roster or tabular form
 In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.
{1, 2, 3, 6, 7, 14, 21, 42}
(ii) Set-builder form
In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set { a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V, we write
V = { x : x is a vowel in English alphabet} 

The Empty Set
 A set which does not contain any element is called the empty set or the null set or the void set.
According to this definition, B is an empty set while A is not an empty set. The empty set is denoted by the symbol φ or { }.
Finite and Infinite Sets
A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite. 
Equal Sets
Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B. 
Subsets
A set A is said to be a subset of a set B if every element of A is also an element of B.
In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “⇒” which means implies.  
Power Set
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
Thus, as in above, if A = { 1, 2 }, then
P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}
Also, note that n [ P (A) ] = 4 = 22
In general, if A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2 m. 
Universal Set 
we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “ Universal Set”. 

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