Operations on sets





Operations on sets can be done by combining two or more sets in different ways to produce another set. Let us discuss the basic operations here:

Union of sets:

For two sets A and B, the operation union is defined as,the set which are containing all the elements of set A and set B, without repeating elements.

The symbol for union is '∪'.

Symbolically we write A union B as A∪B and the definition is written in Roster form as,

A∪B={x:x∈A or x∈B}.

Examples:

  • 1.Find the union of the sets

    A={x:x∈N;5≤x≤10}

    B={x:x=2n, 5≤x≤10, for x∈N}

  • Solution:

    A={5,6,7,8,9,10}

    B={6,8,10}

    So,A∪B={5,6,7,8,9,10}


  • 2. For the sets, P={a,e,o}and Q={i,o,u}, find P∪Q.

  • Solution:

    P={a,e,o}

    Q={i,o,u}

    so,P∪Q = {a,e,i,o,u}

  • Note:

    In both the examples, the elements are not repeated. That is the common elements are written only once.


Intersection of sets:

In the topic operations on sets next we are going to see intersection of two sets.For sets A and B, the operation intersection is defined as, the set which contains the common elements in both A and B.

The symbol to denote the intersection is '∩'.

Symbolically we write A intersection B as, A∩B and the definition for intersection is defined in Roster form as,

A∩B={x:x∈A, and x∈B}

.

Examples:

  • 1. Find the intersection of the given two sets.A={5,7,9,11,13} and B= {5,11,13,15}.
  • Solution:

    A={5,7,9,11,13}

    B={5,11,13,15}

    common elements in A and B are 5, 11 and 13.

    So,A∩B={5,11,13}


  • 2. If P={x:x is an alphabet in the word 'MAY'} and Q={x:x is an alphabet in the word 'JUNE'},then find P∩Q.

  • Solution:

    P={M,A,Y}

    Q={J,U,N,E}

    There is no common elements between P and Q.

    So, P∩Q={}.

Difference of sets:

For sets A and B,

  • A difference B, which is symbolically written as A-B is defined as, A set which contains only the elements of A not the elements of B .

    In Roster form A-B={x:X∈A and x∉B}

  • B difference A, which is symbolically written as B-A is defined as,,A set which contains only elements of B not the elements of A.

    In Roster form B-A={x:x∉B and x∉A}

Examples:

  • 1.If set A={1,3,5} and B={5,6,7,8}, find A-B and B-A.

  • Solution:

    A={1,3,5}

    B={5,6,7,8}

    5 is the common element in A and B

    so,A-B={1,3} and B-A={6,7,8}.


  • 2. For the given sets P={a,e,i} and Q={b,c,d} find P-Q and Q-P.
  • Solution:

    P={a.e,i}

    Q={b,c,d}

    P and Q are disjoint sets, as there is no common element between them.

    So, P-Q={a,e,i}=P it self.

    Q-P={b,c,d}=Q it self

    .

    Complement of a set:

    Let U be the universal set and A is a subset of U. Then the complement of set A, which is denoted as A'is set all elements which are not elements of A but elements in the set U.

    Example:

    • 1.If U={1,2,3,4,5,6,7,8,9,10} and A={2,3,4,8,9,10}. Find A'.

    • Solution:

      U={1,2,3,4,5,6,7,8,9,10}

      A={2,3,4,8,9,10}

      For A' we have to write the elements of the set U which are not elements of A.

      So, A'={1,5,6,7}


    • 2. Find the complement of set B=set of all even natural numbers where, U= set of all natural numbers.

    • Solution:

      U=N

      A={2,4,6,8,....}

      So, A'={1,3,5,7,9,..}=Set of all odd natural numbers.

    Note:

  • Complement of a universal set is the empty set.
  • A set and its complement are always disjoint sets.
  • These are the important topics in operations on sets
    Related Topics


             Math dictionary