## Set Operations

Subject: Compulsory Maths

#### Overview

There are four fundamental or basic set operations. This has the information about the set operation.
##### Set Operations

There are four fundamental or basic set operations. They are given below:

• Union of sets
• Intersection of sets
• Difference of sets
• Complement of a set

1. Union of sets

The union of any two sets A and B is the set of all elements belonging either to A or to B or to both. It is denoted by (AUB) which is read as 'A union B'. The symbol 'U' (cup) is used to denote the union of sets. For example,
A = {a, b, c, d, e}
B = {a, e, i, o, u}
AUB = {a, b, c, d, e} U {a, e, i, o, u}
= {a, b, c, d, e, i, o, u}

2. Intersection of sets

The intersection of any two sets A and B is the set of all elements of both A and B. It is the set of all elements of both A and B. It is denoted by (A∩B) which is read as 'An intersection B'. The symbol '∩' (cup) is used to denote the intersection of sets. For examples:
A = {a, b, c, d, e}
B = {a, e, i, o, u}
A∩B = {a, b, c, e, d}∩ {a, e, i, o, u}
= (a, e}

3. Different of sets

Two sets A and B is called different if the sets. of all the elements that belong to A does not belong to B. It can be written as(A - B). For examples:
A = (a, b, c, d, e}
B = {a, e, i,o, u}
A - B = {a, b, c, d, e} - {a, e, i, o, u}
= {b, c, d}
B - A = {a, b, c, d, e} - {a, e, i, o, u}
= {i, o, u}

4. Complement of a sets
If 'U' be the universal set and A is its subset, then the complement to A is the set of all elements that belong to 'U' but not to A. It can be written as A' or $\overline{A}$ or Ac. For example:
U = {a, b, c, d, e, f, g, h}
A = {a, b, c, d, e}
$\overline{A}$ = U - A
= {a, b, c, d, e, f, g, h} - {a, b, c, d, e}
= {f, g, h}

##### Things to remember
1. There are four fundamental or basic set operations. They are Union of sets, Intersection of sets, Difference of sets, and Complement of a set.
2. The intersection of any two sets A and B is the set of all elements of both A and B.
3. Two sets A and B is called different if the sets. of all the elements that belong to A does not belong to B. It can be written as(A - B).
4. If 'U' be the universal set and A is its subset, then the complement to A is the set of all elements that belong to 'U' but not to A.
5. The union of any two sets A and B is the set of all elements belonging either to A or to B or to both. It is denoted by (AUB)

• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.
##### Sets: Union, Intersection, Complement

Solution:

Here,

P = {a, e, i, o, u}
Q = {a, b, c, d, e}
Now,
P∪Q = {a, b, c, d, e, i, o, u}
The shahded region represents the elements of P∪Q.

P∩Q = {a, e}
The shahded region represents the elements of P∩Q

P - Q = {i, o, u}
The shahded region represents the elements of P - Q

Q - P = {b, c, d}
The shahded region represents the Q - P

Solution:

Here,
U = {1, 2, 3, . . . . . . 15}
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5, 6, 7
Now,
A∪B = {1, 2, 3, 4, 5, 6, 7, 8, 10}
The shaded region represents the elements of A∪B

A∩B = {2, 4, 6}
The shaded region represents the elements of A∩B

A - B = {8, 10}
The shaded region represents the elements of A - B

B - A = {1, 3, 5, 7}
The shaded region represents the elements of B - A

(A∪B)'  = U - (A∪B)
= {1, 2, 3, . . . . . . 15} -  {1, 2, 3, 4, 5, 6, 7, 8, 10}
= {9, 11, 12, 13, 14, 15}
The shaded region represents the elements of (A∪B)'

(A∩B)' = U - (A∩B)
=  {1, 2, 3, . . . . . . 15} -  {2, 4, 6}
= {1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15}
The shaded region represents the elements of (A∩B)'

Solution:

Here,
A∪B = {1, 2, 3, 4, 5, 6, 7}
A∩B = {2, 3, 5}
A - B = {1, 4, 6}
B - A = {7}
(A∪B}' = {8, 9, 10}
(A∩B)' = {1, 4, 6, 7, 8, 9, 10}
(A - B)' = {2, 3, 5, 7, 8, 9, 10}
(B - A)' = {1, 2, 3, 4, 5, 6, 8, 9, 10}

Solution:

Here,
U = {1, 2, 3, . . . . . . . . .  20}
A = {1, 3, 5, 7, 9, 11, 13, 15}
B = {3, 6, 9, 12, 15, 18}
C = {1, 2, 3, 4, 5, 6, 7, 8}
Now,
A∪B = {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18}
∴ A∪B∪C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11., 12, 13, 15, 18}

A∩B = {3, 9, 15}
∴ A∩B∩C = {3, 9}

A∪B = {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18}
∴ (A∪B)∩C = {1, 3, 5, 6, 7, 9}

A∩B = {3, 9, 15}
∴ (A∩B)∪C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 15}

A∪B∪C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 18}
∴ (A∪B∪C)' = U - {A∪B∪C}
= {10, 14, 16, 17, 19, 20}

A∩B∩C = {3, 9}
∴ (A∩B∩C)' = U - {A∩B∩C}
= {1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Solution:

Here,
A = {2, 4, 6, 8, 10, 12}
B = {1, 2, 3, 4, 5, 6}
C = {2, 3, 5, 7, 11}
Now,
B∪C = {1, 2, 3, 4, 5, 6, 7, 11}
∴ A∪(B∪C) = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12}
Again,
A∪B = {1, 2, 3, 4, 5, 6, 8, 10, 12}
∴ (A∪B)∪C = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12}
∴ A∪(B∪C) = (A∪B)∪C proved.

Solution:

Here,
From the Venn diagram
A = {1, 2, 3}
B = {2, 5, 8, 9}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Now,
or, A - B = {1, 2, 3} - {2, 5, 8, 9}
or, A - B = {1, 3}
or, $\overline{A - B}$ = U - (A - B)
or, $\overline{A - B}$ = {1, 2, 3, 4, 5, 6, 7,. 8, 9} - {1, 3}
∴  $\overline{A - B}$ = {2, 4, 5, 6, 7, 8, 9}
In venn diagram

Here, the shaded region represents the set  $\overline{A - B}$

Again,
or, A ∩ B = {2}
or, $\overline{A∩B}$ = U - (A ∩ B)
or, $\overline{A∩B}$ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2}
or, $\overline{A∩B}$ = {1, 3, 4, 5, 6, 7, 8, 9}
In venn diagram,

Here, the shaded region represents the set $\overline{A∩B}$

Then,
or, $\overline{A}$ = U - A
or, $\overline{A}$ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3}
or, $\overline{A}$ = {4, 5, 6, 7, 8, 9}
Now, $\overline{A}$ ∪ B = {4, 5, 6, 7, 8, 9} ∪ {2, 5, 8, 9}
∴ $\overline{A}$ ∪ B = {2, 4, 5, 6, 7, 8, 9}
In venn diagram

Here, the shaded region represents the set $\overline{A}$ ∪ B

And,
or, A∪B = {1, 2, 3} ∪ {2, 5, 9, 8}
or, A∪B = {1, 2, 3, 5, 8, 9}
Now, $\overline{A∪B}$ = U - (A∪B)
or, $\overline{A∪B}\ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 5, 8, 9} or, \(\overline{A∪B}\ = {4, 6, 7} In venn diagram Here, the shaded region represents the \(\overline{A∪B}\ Solution: Here, \(\overline{A∪B}$ = {4, 6, 7}
$\overline{A}$ ∩ $\overline{B}$ = {4, 5, 6, 7, 8, 9} ∩ {1, 3, 4, 6, 7}
$\overline{A}$ ∩ $\overline{B}$ = {4, 6, 7}

Again,
$\overline{A∩B}$ = {1, 3, 4, 5, 6, 7, 8, 9}
$\overline{A}$ ∪ $\overline{B}$ = {4, 5, 6, 7, 8, 9} ∪ {1, 3, 4, 6, 7}
$\overline{A}$ ∪ $\overline{B}$ = {1, 3, 4, 5, 6, 7, 8, 9}

Solution:

Here, from given Venn diagram
U = {a, b, c, d, e, f, g, h, i}
X = {c, f, g}
Y = {c, d, e, f, i}
Z = {b, c, i}

Now,
X∪Y = {g, f, c} ∪ {c, d, e, f,  i}
X∪Y = {c, d, e, f, g, i}
In venn diagram,

Here, the shaded region represents X∪Y

Again,
X ∪ Y ∪ Z = {g, f, c} ∪ {c, d, e, f, i} ∪ {c, i, b}
X ∪ Y ∪ Z = {b, c, d, e, f, g, i}
In venn diagram,

Here, the shaded region represents X ∪ Y ∪ Z.

Similarly,
X ∩ Y = {g, f, c} ∩ {d, f, c, e, i}
X ∩ Y = {c, f}
In venn diagram

Here, the shaded region represents X ∩ Y.

Similarly,
(X ∩ Y) ∩ Z = ({c, f, g} ∩ n{c, d, e, f, i} ∩  {b, c, i}
(X ∩ Y) ∩ Z = {f, c} ∩ {b, c, i}
(X ∩ Y) ∩ Z = {c}
In venn diagram

Here, the shaded region represents (X ∩ Y) ∩ Z

Similarly,
X - (Y ∪ Z) = {g, f, c} - ({c, d, e, f, i} ∪ {c, i, b})
X - (Y ∪ Z) = {g, f, c} - {b, c, d, e, f, i}
X - (Y ∪ Z) = {g}
In venn diagram,

Here, the shaded region represents X - (Y ∪ Z)

Similarly,
X - (Y ∪ Z) = {g, f, c} - ({c, d, e, f, i} ∩ {b, c, i})
X - (Y ∪ Z) = {g, f, c} - {c, i}
X - (Y ∪ Z) = {f, g}
In venn diagram,

Here, the shaded region represents X - (Y ∪ Z)

Solution:

Here, P = {a,e,i,o,u} and Q = {a,b,c,d,e}

Now, P ∪ Q = {a,b,c,d,e,i,o,u}

The shaded region represents the elements of P ∪ Q.

Solution:

Here, P = {a,e,i,o,u} and Q = {a,b,c,d,e}

Now, P - Q = {i,o,u}

The shaded region represents the elements of P - Q.

Solution:

Here, $\overline{A∪B}$ = U - (A∪B)

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

= {9,11,12,13,14,15}

The shaded region represents the elements of $\overline{A∪B}$.

Solution:

Here, $\overline{A-B}$ = U - (A - B)

= U - (8,10)

= {1,2,3,4,5,6,7,9,11,12,13,14,15}

The shaded region represents the elements of $\overline{A-B}$.