SETS AND THEIR OPERATIONS

 we already discuss about set and their types in privious blog , now we discuss on operations on sets.

Operations on set 

1.   Complement of a set :

          Let U is a universal set  and A is a any subset of U then the complememt of A is the elemets is belong in U but not in A.   A complement of of A set is denoted by A' or Ac .                                                It is defined as A' = {x/x∈ U , x ∉ A} = set of all element in U which are not in A .

eg.,  Let X = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, } be the universal set and A ={1,2, 4, 6, 7,}                                                    ∴ The complememt of the set is                                                                                                                          A' = {0, 3,5, 8, 9}

Properties :

i)   (A')' = A      ii)  ф' = U       iii) ሀ' = ф


2.   Union Of Set :

         The union of two set is the set of all element which belong  in  set A or and B . And it is denoted by A ሀ B  . And is defined as  A ሀ B = { x/x ∈ or  x ∈ B }.

eg.,  A = {x/x is a prime number less than 10}and                                                                                               B = {x/x ∈ N , x is factor of 8 }   find A ሀ B .

solution :  Let A = {2, 3, 5, 7}   and B = {1, 2, 4, 8}                                                                                                     A ሀ B  = {1, 2, 3, 4, 5, 7, 8}

Properties of Union set :

  1. A ሀ B = B ሀ A                         ....  (Commutative property)
  2. (A ሀ B ) ሀ C = A ሀ( B ሀ C)     ...  (Associative property )
  3. Aሀф   =  A                               ...  (Identity)
  4. (A ሀ A ) = A                            ...  (Idempotent law )
  5. If A ሀ A'  = U
  6. If  A ⊂ B then A ሀ B = B
  7. U ሀ A = U
  8. A ⊂ (A ሀ B) , B ⊂ (A ሀ B)
3.    Intersection of Set :
 
           The intersection of two set A and ab is the set of all elements which belong in both the set A and B , and is denoted by A⋂B. And it is defined as A⋂B = {x / x ∈ A and x ∈ B}.
eg.,   A = {1,  3, 5, 7, 9, 11, 13}  and B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

        ∴  A⋂B  =  {1, 3, 5, 7, 9} 

Properties of Intersection set :

  1. A ⋂ B = B⋂ A                         ....  (Commutative property)
  2. (A⋂ B )⋂ C = A ⋂( B ⋂C)     ...  (Associative property )
  3. A⋂ф   =  ф                              ...  (Identity)
  4. (A⋂A ) = A                            ...  (Idempotent law )
  5. If A ⋂ A'  =  ф
  6. If  A ⊂ B then A ⋂B = A
  7. U ⋂ A = A                             ....  (Identity for intersection)
  8.  (A ⋂B) ⊂ A , (A ⋂B) ⊂ B
  9.    i)     A ⋂(B ሀ C)  =  (A ⋂B ) ⋃ (A ⋂C)                                                                                              ii)    A ሀ (B ⋂ C)  =  (AሀB)⋂ (AሀC)  .........   (  Distributive law )

4.   Difference of Sets :

     The difference of two set is the set of elements which belongs in set A but not in B and is denoted by A- B  or , A ⋂ B'. Ans is defined as  A- B = { x/x∈ A , x ∉ B} and similary B-A = { y/y∈ B , y ∉ A}

Note : 
  1.  A-B ⊆  A and  B-A ⊆ B.
  2. The sets A-B , A ∩ B and B-A are mutually disjoint sets, i.e. the intersection of any of these two sets is the null set i.e. empty set .
  3. A-B =A ⋂ B'  , B-A = A' ⋂ B 
  4. A⋃ B = (A-B) ሀ (A ∩ B) ሀ (B-A)
  5. (A-B) ሀ (B-A) = A Δ B is called symmetric difference of sets A and B .
Properties :

  1. A Δ B  = (A⋃ B)-(A ∩ B)
  2. A Δ A = ф
  3. A Δ ф = A
  4. If A Δ B = A Δ C then B= C
  5. A Δ B = B Δ A
  6. A ∩ (B Δ C)  =  (A ∩ B) Δ (A ∩ B)

Properties of Cardinality of Sets :

  • n(A⋃ B) = n(A) + n(B) -  n(A ∩ B)
  • When A and B are disjoint sets, then n(A⋃ B) = n(A) + n(B)  , as A ∩ B =ф , n(A ∩ B) =0 
  • n(A ∩ B') + n (A ∩ B) = n(A)
  • n(A' ∩ B) + n (A ∩ B) = n(B)
  • n(A ∩ B') + n (A ∩ B) + n(A' ∩ B)= n(A⋃ B)
  • For any sets A ,B and C                                                                                                                      n (A ሀ B ሀ C) = n (A) +n(B) + n(C) - n(A ∩ B) - n(B ∩ C) -n (A ∩ C) + n (A ∩ B∩ C)
  • If n(A) = m  n[P(A)] = 2m ,where P(A) is a power set of A .
  • n (A Δ B ) =n(A) + n(B) -2n (A ∩ B).

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