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Sets

A set is a well-defined collection of objects. A collection is said to be well-defined if it is possible to say clearly whether a particular object belongs to it or does not belong to it. Objects can be anything ranging from living beings to non-living things. It may be a group of people, a collection of letters or numbers or even a collection of things which we encounter in everyday life.


Each object is called an element of the set. Usually sets are denoted by a capital letter like A, B, C, D, etc.

Methods of Representing a Set

  • Roster form (tabulation method or algebraic form): In this method all the elements of the set are listed within curly brackets ‘{}’ and separated by a comma.
     

    Example: Let A be a set of vowels. Then, A = {a, e, i, o, u}.

     

  • Set builder form (rule form): In this method, a set is described by the characterizing property of its elements. In this method, we first write a variable ‘x’ within the curly brackets that stands for all the variables of the set followed by the property that governs the elements of the set. The symbol ‘:’ is used between the variable and the property which means ‘such that’. This method of writing the set is called property method.
     

    Example: Let A be a set of vowels. In the set builder form, it is shown as A = {xx is a vowel}.

Note: The order of the elements in a set is not relevant. Two identical elements are considered to be one element. Repetition of elements in a set is meaningless.

Different Types of Sets

  • Null set: The set which has no elements in it is known as a null set.
     
    Null set is denoted by Φ, where Φ = {}
     

    Example: A = {xx is an odd number divisible by 2} is a null set as there are no odd numbers that are divisible by 2.

     

  • Singleton set: A set containing only one element is known as a singleton set.
     

    Example: A = {xx is the present capital city of India} is a singleton set as there is only one element.

     

  • Universal set: A set which has all possible elements of a situation or problem under consideration is known as a universal set. Universal set is denoted by m or U.
     

    Example: The set of all natural numbers is a universal set.

  • Subset: A set ‘A’ will be a subset of ‘B’, if all elements of A are present in B. It is denoted by A  B.
     

    Example: Let B = {1, 2, 3, 4, 5} and A = {1, 2, 3, 4, 5}

     

    Then A is called a subset of B or A  B.

  • ƒProper subset: A set A will be a proper subset of B, if all elements of A are there in B, but at least one element of B is not in A or we can say when A and B are not equal. Proper subset is denoted by A  B.
     

    Example: Let B = {1, 2, 3, 4, 5} and A = {1, 2, 3}
    Here, A is called the proper subset of B or A  B.

Note: If a set has n elements in it, then it can have 2n number of subsets and 2n - 1 number of proper subsets.

 

     Null set and the set itself are considered as subsets for every set.

  • ƒSuperset: If set A is a proper subset of B, then B is called the superset of A and is denoted by B  A.
     

    Example: Let B = {1, 2, 3, 4, 5} and A = {1, 2, 3}.

     

    Here, A is called the proper subset of B and B is called the superset of A or B  A.

Note: Subset can also be represented by the symbol .

  • Power set: The set of all subsets of a set A is called power set of A. It is denoted by P(A).
     

    Example: Let A = {1, 2, 3}. Subsets of A are {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}.

     

    Then, set of all subsets P (A) = {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}.

  • Equal sets: Two sets are said to be equal sets, if each and every element in both the sets is same.
     

    Example: Let A = {a, b, c, d, e} and B = {b, c, a, e, d}

     

    Then, A and B are equal sets.

     

  • Equivalent sets: Two sets are said to be equivalent sets, if they have the same number of elements.
     

    Example: Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5}

     

    Then, A and B are equivalent sets.

Note: All equal sets are equivalent, but all equivalent sets need not be equal.

Venn Diagram

Venn diagram is a symbolic representation of a set.
 

The universal set is shown by a rectangular region and each of its subsets is shown by circular region.

 

Example: Let N be a set of all natural numbers. Consider a set A = {1, 2, 3, 4, 5}
This can be represented on a Venn diagram as
Description: 57487.png 
Here N is the universal set and A is a subset of N.

Operations on a Set

  • Union set: Let A and B be two sets. Then set of elements of both A and B combined, is said to be the union set of A and B. It is denoted by A  B.
     
    Description: 62918.png 

Example: A = {a, b, e}, B = {c, d, e}
∴ A ∪ B = {a, b, c, d, e}
A ∩ B = { } = Ø

  • Intersection set: Let A and B be two sets. Then the set of elements which are in A as well as in B is said to be the intersection set of A and B and it is denoted by A  B.
     

    Example: A = {a, b, c, d}, B = {c, d, e}

     
     A  B = {c, d}
     

    Description: 62999.png

  • Disjoint sets: Let A and B be two sets. If there are no common elements in sets A and B, then the sets are said to be dis-joint sets.
     

    Example: A = {1, 2}, B = {x, y}

     
     A  B = { } = Ø
     

    Description: 63006.png

  • Difference set: Let A and B be two sets. The set of elements in A which is not present in B is called difference set of A and B and it is denoted by A – B.
     
    Description: 62939.png Description: 62932.png 
     

     

    Example: A = {1, 2, 3, 4, 5, 6}, B = {5, 6, 7, 8, 9}

     

     A - B = {1, 2, 3, 4}    and    B - A = {7, 8, 9}

  • Complementary set: Let U be the universal set and A be any non-null set. The set of elements which are in U but not in A is said to be the complementary set of A and it is denoted by A′ or AC.
     

    Example: U = {a, b, c, d, e, f }, A = {c, d, e}

     
     A′ = U - A = {a, b, f }
     

    Description: 63013.png

Note: A  A′ = U and A  A′ = 

Laws of Set Operations

  • Commutative Law: If A and B are any two sets, then
     
    (i) A  B = B  A
     
    (ii) A  B = B  A
  • Associative Law: If AB and C are any three sets, then associative law is said to hold good.
     
    (i) (A  B)  C = A  (B  C).
     
    (ii) (A  B)  C = A  (B  C).
  • Distributive Law: If A, B and C are three sets and  and  are the operations, then
     
    (i) A  (B  C) = (A  B)  (A  C).
     
    (ii) A  (B  C) = (A  B)  (A  C)
  • ƒIdempotent Law: If A is any set, then
     
    (i) A  A = A
     
    (ii) A  A = A
  • Identity Law: If A is any set and U is the universal set and φ is the null set, then
     
    (i) A  φ = A
     
    (ii) A  U = A
  • Complement Law: If A is any set and A′ is the complement of the set A, then
     
    (i) A  A = φ
     
    (ii) A  A = U.
  • De Morgan’s Law: If A and B are any two sets, then
     
    (i) (A  B) = A  B.
     
    (ii) (A  B) = A  B.

Cardinal Numbers

The number of elements in a set is known as its cardinal number.

 

The cardinal number of a set A is denoted as n(A).

 

Example: Consider a set A = {1, 2, 3, 4, 5, 6}, then n(A) = 6.

 

Important results:

  • ƒn(A  B) = n(A) + n(B) - n(A  B)
  • If A  B = f, then n(A  B) = n(A) + n(B)
  • n(A - B) + n(A  B) = n(A)
  • n(B - A) + n(A  B) = n(B)
  • n(A - B) + n(A  B) + n(B - A) = n(A  B)
  • n(A  B  C) = n(A) + n(B) + n(C) - n(A  B) - n(B  C) - n(C  A) + n(A  B  C)
     
    n’ represents the number of elements in the set given within the brackets.

Note:

  • If A contains ‘n’ different elements, then P(A) contains ‘2n’ different elements
  • In two sets, if one is the subset of other, then the sets are called comparable sets

 

Example
In a survey of 50 students, it was found that 30 had taken up accounts, 24 had taken up economics, 22 had taken up law, 10 had taken up accounts and law, 18 had taken up accounts and economics, 8 had taken up economics and law and 6 had taken all the three subjects.
Find the number of students who had taken up
i. Only law
ii. Only accounts
iii. Only economics
iv. Economics and law but not accounts
v. Accounts and economics but not law
vi. Only one of the subjects
Solution
n(A) = 30
 
n(E) = 24
 
n(L) = 22
 
n(A ∩ L) = 10
 
n(A ∩ E) = 18
 
n(E ∩ L) = 8
 
n(A ∩ E ∩ L) = 6
 
From the Venn diagram, we can see that
 
Description: 62971.png 
 
  1. Number of students who have taken up only law = n(L) - n(A ∩ L) - n(L ∩ E) + n (A ∩ E ∩ L= 22 - 10 - 8 + 6 = 10
  2. Number of students who have taken up only accounts = n(A) - n(A ∩ L) - n(A ∩ E) + n(A ∩ E ∩ L) = 30 - 10 - 18 + 6 = 8
  3. Number of students who have taken up only economics = n(E) - n(A ∩ E) - n(L ∩ E) + n(A ∩ E ∩ L= 24 - 18 - 8 + 6 = 4
  4. Number of students who have taken up economics and law but not accounts = n(E ∩ L) - n(A ∩ E ∩ L) = 8 - 6 = 2
  5. Number of students who have taken up accounts and economics but not law = n(A ∩ E) - n(A ∩ E ∩ L) = 18 - 6 = 12
  6. Number of students who have taken up only one of the subjects = Number of students who have taken up only law + Number of students who have taken up only accounts + Number of students who have taken up only economics = 10 + 8 + 4 = 22.

Cartesian Product of Sets

Let A and B be two sets. Then Cartesian product of A and B is denoted by A × B and is defined as under:

 

A × B = {(a, b) : a  A and b  B}

 

n(A × B) = n(A) × n(B)

 

For example, A = {a, b, c}, B = {x, y}

 

 A × B = {(a, x), (a, y), (b, x), (b, y), (c, x), (c, y)}

Results of Cartesian Products of Sets

  1. A × B ≠ B × A
  2. A ×  =  × A
  3.  ×  = 
  4. A × (B  C ) = (A × B (A × C )
  5. A × (B  C ) = (A × B (A × C )
  6. A × (B - C ) = (A × B) - (A × C )
  7. n(A × B) = n(A) × n(B)

Example
Description: 54183.png
Solution
{1, 2, 5}, B = { 1, 5 } A ∩ B = {1, 5}

 

Example
If A = {3, 4, 5}, then n(A) = ?
Solution
Here, n(A) = number of different elements in A = 3

 
Example
If A = {5, 6, 7}, B = {6, 7, 8, 3}, then A - B = ?
Solution
A - B = {5}

 
Example
If n(A) = 3, then n[P(A)] = ?
Solution
n(P(A)) = 2n(A) = 23 = 8

 

Example
If A = {a, b}, B = {c, d}, then A × B = ?
Solution
A × B = {(a, c), (a, d), (b, c), (b, d)}

 

Example
Description: 54195.png
Solution
Description: 56324.png

 
Example
Description: 56338.png
Solution
Description: 56346.png

 

Example
Description: 56352.png
Solution
Description: 56361.png
= {1, 2, 3, 4, 5, 6, 7, 8} – {1, 2, 3, 4, 5, 6} = {7, 8}

 

Example
Description: 56367.png
Solution
Description: 56373.png
Description: 56379.png





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