Set And Functions

Set And Functions

Sets

   A Set is a collection of well-defined objects. Here, 'well-defined' means that there is a definite method to determine whether an objects belongs to the set. The members of a set are called the elements of the set . Sets are usually denoted by capital letters of the English alphabets while the elements of the denoted by small letters of English alphabet. For example , We may define A to be set of odd numbers . Recall that an integer is called an odd number, if it is divisible by 2. Mathematically , A = {x | x is an integer }
  • If x is an element of the set A , then we write x ∈ A.
  • If x is not an element of the set A, Then we write x ∉ A

Representation of Sets 

Sets are usually described into ways:

1. Tabular form or roster form:- In this form, all the elements of the set are separated by commas and enclosed between the bracket {}.
  e.g. (i) The set of vowels of English alphabets as A = {a,e,i,o,u}.
         (ii) The set of numbers on a clock face is written as 
                            B = {1,2,3,4,5,6,7,8,9,10,11,12}.

2. Set builder form :- We define a set by stating some properties, which its elements must satisfy. 
  e.g. The set of all even integers, then we use the letters usually x and write 
                           A = {x | x is an even integer}
   This is to be read as A is a set of numbers x such that x is an even integer. The vertical line '|' to be read as such that some times we use ':' in place of vertical line.
  e.g.                   A = {x:x is an even integer}
                 and    C = {1,w,w2} = {x | x3 - 1 = 0}

Types of sets

1.  Empty or null set :-   The set, which contains no element, is called an empty set or null set and it is denoted by 'Φ' i.e. Φ = {} (as there is no element in this set). The null set is Φ is  subset of every set A.

e.g. The set of odd numbers divisible by 2 is a null set.

2. Singleton set :- A set contain only one element is called a singleton set .
   e.g. {1} and {4} are singleton sets`

3 Equality of sets :- The sets A and B are said to be equal, if  they have same members i.e. if every element of A is an element of B and every element of B is an element of A , then A = B

 e.g. If   A = {1,3,5,7}
             B = {7,3,1,5}
         then A = B
** If the two sets are not equal , we write A ≠ B. A set does not change , if its elements are repeated or if the order of its elements id different.

4. Finite Sets :-  The set , which contains a definite (finite) number of elements , is called a finite set.
  e.g. The set of days in a week.

5. Infinite sets :- The set , which contains an infinite number of elements, is called an infinite set.
         e.g. The set, of numbers of people on Earth.

6. Disjoint sets :- Two sets A and B are said to be disjoint , if they do not have any element in common .
   e.g. A = {1,2,3} and B ={4,5,6} are disjoint.

7. Subset :- If every element of set A is also an element of another set B, then A is called a subset of B , Also, B is said to be super set of A.
 Symbolically, we write
                       A  ⊆ B (if A is contained in B)
and                B ⊇ A (if B is contains A)

 More specifically we say, A ⊆ B if x ∈ A
                                     x ∈ A
e.g. A ⊆ B since every element of A is in B.
If there is atleast one element of A , which is not in B, then A is not a subset of B written as 
A ⊆  B.

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