SET THEORY

Set – 

Set is a collection of well- defined objects which are distinct from each other. 

For example –

                         (1) A = { a ,b, c } 

                         (2)set of first ten natural numbers { 1,2,3,….10 } 

                         (3)set of all rivers in India . 

Representation of Sets – 

1. Roster /listing method/tabular form – 

In this method, a set is described by listing elements, separated by commas, within braces { }. 

For example … A = { a , e , i, o , u } 

 

1. Set builder method – 

In this method, we write down a property or rule which gives us all the elements of the set by that rule.  

For example – A = { x: x is a vowel of English alphabets .} 

 

3.    Notation of Set – 

Sets are generally denoted by capital letters such as A, B, C, X, Y, Z, etc. 

 

4.     Elements of a Set – 

An element ( or member ) of a set is any one of the distinct objects that make up that set. 

* They are denoted by small letters a, b,c x,y,z, etc. 

*If x is an element of a set A, then it is denoted by x ∈ A. 

*If x does not belong to a set A, then it is denoted by x ∉ A. 

*Number of elements of a set is denoted by n(A) and it is the number of objects in that set. 

 

Some Important Sets of Numbers --- 

1. The set N = { 1, 2 ,3, ……} of all natural numbers . 

        2. The set I or Z = { …., -3 ,-2 , -1 , 0 , 1 ,2 ,3 , ….} of integers . 

        3.The set Q = { x :x = p/q , where p and q are integers and  q ≠ 0 } of all rational numbers . 

        4.The set Q+ of all positive rational numbers. 

        5.The set R of all real numbers . 

         6.The set R+ of all positive real numbers . 

          7.The set C of all complex numbers . 

 

Examples --- 

1. Express the given set in roster form 

A = { x :x is an integer and 5

 

Sol. The integers between 5 and 9 are 6,7,8. 

Thus A = { 6 , 7 , 8 } 

       2.    Express the given set in set builder form  

             A = { 1,4,9,16,25,….} 

             Sol.  The given set in set builder form is  

             A = { x :x is square of a natural number }. 

Types of Sets --- 

 

(1)Null set or empty set

       If the set has no elements, it is called an empty or null or void set. 

       It is denoted by { } or ∅ . 

For example 

      A = { x: x² = 16 , x is an odd number } is an empty set because 16 is not a square of any odd number.        

(2)Singleton set

      The set consisting of only one element is known as a singleton set. 

For example 

      A = { x: x +5 = 9 } is a singleton set . 

(3)Finite and Infinite sets 

     A set will be called finite if either it is empty or has a finite number of elements. 

     A set with an infinite number of elements is referred to as an infinite set. 

For example

    The set { 1,2,3,…….,1000 } is a finite set, whereas the set of all integers is infinite. 

(4)Equal sets

     If A, B are set such that every element of A belongs to B, and every element of B belongs to A, then A and B are the same sets and we write  

             A = B 

For example

     A = { 3,5,7 }    , B = { 7,5,3 } 

Here, A =B 

(5)Power set (P)

      The set formed by all the subsets of a given set A is called power set of A, denoted by P(A). 

     Also if any set A contains n elements, then P(A) contains 2ⁿ elements. 

For example

     A = { 1,2,3 } , then  

     P(A) = [ { } , {1} , { 2} , {3},{1,2} , { 1,3} , {2,3} ,{1,2,3} ] 

(6)Comparable and Non-Comparable sets

    Two sets A and B are comparable, if  

     A ⊆B or B ⊆A  

     For two sets A and B, if neither A ⊆B nor B ⊆A, then A and B are non-comparable sets. 

(7)Subset 

    If A and B are set such that every element of A is also an element of B, then A is said to be a subset of B. 

    Symbolically, we write  

                        A   ⊆B 

    Which is read “A is a subset of B” or “A is contained in B”. 

    Also B is called superset of A or B contains A  

                        B ⊇A  

For example 

    Let B = { 1,2,3,4} and A = { 1,2,3}  

    We see that every element of A is an element of B , then A is a subset of B. 

(8)Proper subset 

    If there is at least one element in B which is not in A , we say A is strictly contained in     B and call A a proper subset of B .Symbolically ,  

                  A ⊂B 

(9)Universal set

    In any application of set theory, all the sets under consideration will likely be subsets of a fixed set. We call such a set the Universal set and it is denoted by U. 

Note – 

    (1)An empty set is a subset of every set. 

    (2) Every set is a subset of itself. 

    (3) If the number of elements in any set is n, then the number of possible subsets for that set is 2ⁿ. 

Illustrative Examples --- 

    (1)If A = { a ,b ,c } , then find the number of possible subsets for A . 

Sol.  Here n(A) = 3 , then number of possible subsets 

                                  = 2³ 

                                  = 8 

    (2)If A = { 1,2,3,4} and B= { 1,2,3,4 } , then A is a proper subset of B or not ? 

Sol.    Here A = B , then A is not a proper subset of B . 

    (3)If A = ∅ , then find P(A) and P(P(A)) . 

Sol.   A = ∅ , then  

P(A) = { ∅ } 

And     P(P(A)) = { ∅ , { ∅ } } 

     (4)Is A = { n: n is an integer and n² < 5 } and B ={x :x is a real number and x² - 3x +2 =0}  are equal sets ? 

Sol.  Here A = { -2,-1,0,1,2 } and B = { 1,2 }  

Thus , A and B are not an equal sets . 

    (5)Is A = { x: x is a real number and x/2 + 7 = 13 } , a singleton set ? 

Sol.   x/2 + 7 = 13 

Or      x = 12 

Thus , A = { 12 }  

Therefore , A is a singleton set .