Complement of a Set

Definition: 

The complement of a set A, is set of all elements of universal set, which are not the elements of A.

Let U be the Universal set and A is subset of U, then the “Complement of A” is set of all elements of U which are not the elements of A.

Complement of a Set A is denoted by A’

              A’ = {x : x ∈ U and x ∉ A}, also 

                          A’ = U – A 

The Complement A’ of a set A can be represented by Venn Diagram in below figure.

The shaded portion shows the complement of set A, this shows that all the elements which are not the elements of A. 

               Complement of a Set Examples

Example 1: U(universal set) is a set of natural numbers and A is a set of odd numbers find A’?

Given a universal set and a set of odd numbers.

     U = {x: x is set of natural numbers} and

     A = {x: x is a set of odd natural numbers}

  ∴ A’ = {x: x is a set of even natural numbers}

Example 2: U(universal set) is a set of integers and A is a set of odd integers find A’?

Given a universal set and a set of odd numbers.

     U = {x: x is set of integers} and

     A = {x: x is a set of odd integers}

  ∴ A’ = {x: x is a set of even integers}

Example 3: Universal set is letters in English alphabet and A is a set of vowels in English alphabet.
find A’?

 Given a universal set and a set of vowel in English alphabet. 

     U = {x: x is set of English alphabet} and

     A = {x: x is a set of vowel in English}

  ∴ A’ = {x: x is a set of consonants}

     

    Complement of a Set with Venn Diagrams

Example 1: If  U = {1, 2, 3, 4, 5,}, A = {1, 2, 3,}  find A’?

Solution: Given that U = {1, 2, 3, 4, 5,} and

                              A = {1, 2, 3,}

We observe that 4 and 5 only elements of universal set which do not belong to A.

                      ∴     A’ = {4, 5,}

           Therefore    A’ = {4, 5,}

Example 2: If U = {a, b, c, d, e,} A = {a, b, c,}  
find A’?  

Solution: Given that U = {a, b, c, d, e,} and

                              A = {a, b, c,}

We observe that only d and e elements of universal 
set which do not belong to A.               

                      ∴      A = {a, b, c,}

           Therefore    A’ = {d, e,}

Complement of union of Set with Venn Diagrams

Example 3: If U = {a, b, c, d, e, f, g}  and (A∪B) = {a, b, f, g}, find (A∪B)’?  

Solution: Given that U = {a, b, c, d, e, f, g} and

                       (A∪B) = {a, b, f, g}

                   ∴  (A∪B)’ = {c, d, e}

             Law of empty set and Universal set

(1) Complement of a universal set is a empty set.

                               U’ =  ∅

(2) Complement of a empty set is a universal set.

                               ∅’ =  U

(3) The set and its complement are disjoint sets.

           

            Properties of Complement of sets

1.          a.  A∪A’ = U   {Law of Complement}
             b.  A∩A’ = ∅    {Law of Complement}

2.          a.  {A∪B}’ = A’∩B’ {De Morgan’s Law}
             b. { A∩B}’ = A’∪B’ {De Morgan’s Law }

3.          {A’}’ = A {Law of double Complementation}

4.          U’ =  ∅  and  
             ∅’ = U {Law of empty set and Universal set}