##
**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*

*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*.

*You can comment your questions or problems regarding the set operations and venn diagrams here.*

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