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

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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}

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}

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}

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 }

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} 

The shaded region represents the difference A – B.

ii.        A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

B – A = {5, 6} 

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 A^c is the set of all the elements of set U which are not in set A.

Therefore, A^c = 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,

A^c = U – A = {1, 3, 5, 7, 9, 10} 

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,

               The shaded region represents the the set (A ∪ B) ∩ C.

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