Set Operations with Venn Diagrams

Set Operations with Venn Diagrams

Set Operations with Venn Diagrams

There are mainly four operations of sets. They are:
i.       Union of sets
ii.      Intersection of sets
iii.     Difference of sets
iv.     Complement of a set

i.  Union of sets

The union of two sets A and B is the set of all elements that belongs to set A or set B or both the set A and B. Symbolically we write ‘A B’ and read as ‘A union B’.

Therefore, A B = {x:x A or x B or x both A and B}

For example:
i.         A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
A B = {1,2,3,4,5,6}
Union of sets A and B
The shaded region represents the union of sets A and B.

ii.        P = {a, b, c, d, e} and Q = {c, d}
P Q = {a,b,c,d,e}
Union of sets P and Q

The shaded region represents the union of sets P and Q.

ii.  Intersection of sets

The intersection of two sets A and B is the set of all common elements that belongs to both the sets A and B. Symbolically, we write ‘A B’ and read as ‘A intersection B’.

Therefore, A B = {x:x A and x B}

For example:
i.         A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
A B = {3, 4}
Intersection of sets A and B

The shaded region represents the intersection of sets A and B.

ii.        P = {a, b, c, d, e} and Q = {c, d}
P Q = { c, d }
Intersection of sets P and Q

The shaded region represents the intersection of sets P and Q.

iii.  Difference of sets

The difference of two sets A and B written as A – B is the set of all elements of set A only which are not in set B. Similarly, the difference B – A is the set of all elements of set B only which are not in set A.

Therefore, A B = {x:x A and x B} and B A = {x:x B and x A}

For example:
i.         A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
A B = {1, 2} 
Difference of sets A – B

The shaded region represents the difference A B.

ii.        A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
B A = {5, 6} 
Difference of sets B – A

The shaded region represents the difference B A.

iv.  Complement of a set

Let U be the universal set and A is any subset of U, then the complement of set A denoted Ac is the set of all the elements of set U which are not in set A.

Therefore, Ac = U – A ={x:x U and x A}

For example:
Let, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8} then,
Ac = U – A = {1, 3, 5, 7, 9, 10} 
Complement of set A

The shaded region represents the complement of A.


Workout Examples

Example 1: If U = {1, 2, 3, ……… 15}, A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8, 10, 12} and C = {3, 6, 9, 12, 15} then find (A B) C and show in the venn diagram.

Solution: Here,
                                U = {1, 2, 3, ……… 15}
                                A = {1, 2, 3, 4, 5, 6}
                                B = {2, 4, 6, 8, 10, 12}
                                C = {3, 6, 9, 12, 15}
                Now,
                                (A B) C = [{1, 2, 3, 4, 5, 6} {2, 4, 6, 8, 10, 12}] {3, 6, 9, 12, 15}
                                                     = {1, 2, 3, 4, 5, 6, 8, 10, 12} {3, 6, 9, 12, 15}
                                                     = {3, 6, 12}
                Venn diagram,
(A ∪ B) ∩ C
               The shaded region represents the the set (A  B)  C.


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