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Set Operations

Subject: Compulsory Maths

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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.
Videos for Set Operations
[Discrete Math 1] Set Operations
Sets: Union, Intersection, Complement
Questions and Answers

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 \(\overline{A-B}\).

Quiz

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