Subject: Compulsory Maths

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

**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}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:**Intersection of sets**

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}**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}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 A**Complement of a sets**^{c}. 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}

- There are four fundamental or basic set operations. They are Union of sets, Intersection of sets, Difference of sets, and Complement of a set.
- The intersection of any two sets A and B is the set of all elements of both A and B.
- 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).
- 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.
- 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.

If P = {a, e , i , o, u} and Q = {a, b, c, d, e}, find P∪Q, P∩Q, P - Q and Q - P.

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

If U = {1, 2, 3, . . . . . . 15}, A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6, 7} form the following stes and represent them in Venn-diagram.

- A∪B
- A∩B
- A - B
- B - A
- (A∪B)'
- (A∩B)'

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)'

From the adjoining Venn-diagram, list the elements of the following sets.

- A∪B
- A∩B
- A - B
- B - A
- (A∪B)'
- (A∩B)'
- (A - B)'
- (B - A)'

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}

If U = {1, 2, 3, . . . . . . . . . 20}, A = {1, 3, 5, 7, 9, 11, 13, 15}, B = {3, 6, 9, 12, 15, 18} and C = {1, 2, 3, 4, 5, 6, 7, 8}, list the elements of the following sets.

- A∪B∪C
- A∩B∩C
- (A∪B)∩C
- (A∩B)∪C
- (A∪B∪C)'
- (A∩B∩C)'

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}

If A = {2, 4, 6, 8, 10, 12}, B = {1, 2, 3, 4, 5, 6} and C = {2, 3, 5, 7, 11}, show that A∪(B∪C) = (A∪B)∪C

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.

On the basis of the given Venn diagram, list each of the following sets and respect each o f them by shading in separate each Venn diagram.

- \(\overline{A - B}\)
- \(\overline{A ∩ B}\)
- \(\overline{A}\) ∪ B
- \(\overline{A∪B}\)

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}\

From the given Venn diagram list the following sets using listing method:

- \(\overline{A∪B}\) and \(\overline{A}\) ∩ \(\overline{B}\)
- \(\overline{A∩B}\) and \(\overline{A}\) ∪ \(\overline{B}\)

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}

Look at the Venn diagram and list each of the sets and represent each of them by shading in separate Venn diagrams.

- X ∪ Y
- X ∪ Y ∪ Z
- X ∩ Y
- (X ∩ Y) ∩ Z
- X - (Y ∪ Z)
- X - (Y ∩ Z)

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)

If P = {a,e,i,o,u} and Q = {a,b,c,d,e} find P ∪ Q.

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.

If P = {a,e,i,o,u} and Q = {a,b,c,d,e} find 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.

If U = {1,2,3,.......15}, A = {2,4,6,8,10} and B = {1,2,3,4,5,6,7} find \(\overline{A∪B}\) and represent in Venn diagram.

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 \(\overline{A∪B}\).

If U = {1,2,3,.......15}, A = {2,4,6,8,10} and B = {1,2,3,4,5,6,7} find \(\overline{A-B}\) and represent in Venn diagram.

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 \(\overline{A-B}\).

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