NCERT Solutions For Class 11 Maths Chapter 2 Exercise 2.2

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Chapter 2 Ex.2.2 Question 1

Let \(A = \left\{ {1,2,3, \ldots ,14} \right\}\). Define a relation \(R\) from \(A\) to \(A\) by \(R = \left\{ {\left( {x,y} \right),3x - y = 0;x,y \in A} \right\}.\) Write down its domain, co-domain and range.

Solution

The relation \(R\) from \(A\) to \(A\) is given as \(R = \left\{ {\left( {x,y} \right),3x - y = 0;x,y \in A} \right\}.\)

Thus, \(R = \left\{ {\left( {x,y} \right),3x = y;x,y \in A} \right\}.\)

Therefore, \(R = \left\{ {\left( {1,3} \right),\left( {2,6} \right),\left( {3,9} \right),\left( {4,12} \right)} \right\}\)

The domain of \(R\) is the set of all first elements of the ordered pairs in the relation.

Hence, Domain of \(R = \left\{ {1,2,3,4} \right\}\)

The whole set \(A\) is the co-domain of the relation \(R.\)

Therefore, Co-domain of \(R = A = \left\{ {1,2,3, \ldots ,14} \right\}\)

The range of \(R\) is the set of all second elements of the ordered pairs in the relation.

Therefore, Range of \(R = \left\{ {3,6,9,12} \right\}\)

Chapter 2 Ex.2.2 Question 2

Define a relation \(R\) on the set \(N\) of natural numbers by \(\rm{R} = \{ \left( {x,{\rm{ }}y} \right):{\rm{ }}y = x{\rm{ }} + {\rm{ 5}},\)  \(x\) is a natural number less than \(4;\;x,\,y \in {\rm{N}}\} .\)

Depict this relationship using roster form. Write down the domain and the range.

Solution

\(\rm{R} = \{ \left( {x,{\rm{ }}y} \right):{\rm{ }}y = x{\rm{ }} + {\rm{ 5}},\)  \(x\) is a natural number less than \(4;\;x,\,y \in {\rm{N}}\} .\)

The natural numbers less than \(4\) are \(1,\; 2,\) and \(3.\)

Therefore, \({\rm{R = }}\left\{ {\left( {{\rm{1}},{\rm{6}}} \right),\left( {{\rm{2}},{\rm{7}}} \right),\left( {{\rm{3}},{\rm{8}}} \right)} \right\}\)

The domain of \(R\) is the set of all first elements of the ordered pairs in the relation.

Hence, Domain of \({\rm{R}} = \left\{ {{\rm{1}},{\rm{2}},{\rm{3}}} \right\}\).

The range of \(R\) is the set of all second elements of the ordered pairs in the relation.

Therefore, Range of \(R = \left\{ {6,7,8} \right\}\).

Chapter 2 Ex.2.2 Question 3

\(A = \left\{ {1,2,3,5} \right\}\) and \(B = \left\{ {4,6,9} \right\}\). Define a relation \(R\) from \(A\) to \(B\) by \(R = \left\{ {\left( {x,y} \right):{\text{the difference between }}x{\text{ and }}y{\text{ is odd}};x \in A,y \in B} \right\}.\) Write \(R\) in roster form.

Solution

\(A = \left\{ {1,2,3,5} \right\}\), \(B = \left\{ {4,6,9} \right\}\) and

\(R = \left\{ {\left( {x,y} \right):{\text{the difference between }}x{\text{ and }}y{\text{ is odd}};x \in A,y \in B} \right\}\)

Therefore, \(R = \left\{ {\left( {1,4} \right),\left( {1,6} \right),\left( {2,9} \right),\left( {3,4} \right),\left( {3,6} \right),\left( {5,4} \right),\left( {5,6} \right)} \right\}\)

Chapter 2 Ex.2.2 Question 4

The Fig2.7 shows a relationship between the sets \(P\) and \(Q.\) Write this relation

(i) in set-builder form

(ii) in roster form.

What is its domain and range?

Solution

As per the Fig2.7, \(P = \left\{ {5,6,7} \right\}\) and \(Q = \left\{ {3,4,5} \right\}\)

(i) \(R = \left\{ {\left( {x,y} \right):y = x - 2;x \in P} \right\}\) or \(R = \left\{ {\left( {x,y} \right):y = x - 2,for{\rm{ }}\;x = 5,6,7} \right\}\)

(ii) \(R = \left\{ {\left( {5,3} \right),\left( {6,4} \right),\left( {7,5} \right)} \right\}\)

Domain of \(R = \left\{ {5,6,7} \right\}\)

Range of \(R = \left\{ {3,4,5} \right\}\)

Chapter 2 Ex.2.2 Question 5

Let \(A = \left\{ {1,2,3,4,6} \right\}\). Let \(R\) be the relation on \(A\) defined by

\(\left\{ {\left( {a,b} \right):a,b \in {\rm{A}},b{\text{ is exactly divisible by }}a} \right\}\)

(i) Write \(R\) in roster form

(ii) Find the domain of \(R\)

(iii) Find the range of \(R.\)

Solution

It is given that \(A = \left\{ {1,2,3,4,6} \right\}\) and \(R = \left\{ {\left( {a,b} \right):a,b \in {\rm{A}},b{\text{ is exactly divisible by }}a} \right\}\)

(i) \(R = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {1,6} \right),\left( {2,2} \right),\left( {2,4} \right),\left( {2,6} \right),\left( {3,3} \right),\left( {3,6} \right),\left( {4,4} \right),\left( {6,6} \right)} \right\}\)

(ii) Domain of \(R = \left\{ {1,2,3,4,6} \right\}\)

(iii) Range of \(R = \left\{ {1,2,3,4,6} \right\}\)

Chapter 2 Ex.2.2 Question 6

Determine the domain and range of the relation \(R\) defined by

\(R = \left\{ {\left( {x,x + 5} \right):x \in \left\{ {0,1,2,3,4,5} \right\}} \right\}\).

Solution

It is given that \(R = \left\{ {\left( {x,x + 5} \right):x \in \left\{ {0,1,2,3,4,5} \right\}} \right\}\)

Therefore, \(R = \left\{ {\left( {0,5} \right),\left( {1,6} \right),\left( {2,7} \right),\left( {3,8} \right),\left( {4,9} \right),\left( {5,10} \right)} \right\}\)

Hence,

Domain of \(R = \left\{ {0,1,2,3,4,5} \right\}\)

Range of \(R = \left\{ {5,6,7,8,9,10} \right\}\)

Chapter 2 Ex.2.2 Question 7

Write the relation \(R = \left\{ {\left( {x,{x^3}} \right):x{\text{ is a prime number less than }}10} \right\}\) in roster form.

Solution

It is given that \(R = \left\{ {\left( {x,{x^3}} \right):x{\text{ is a prime number less than }}10} \right\}\)

The prime numbers less than \(10\) are \(2,\; 3,\; 5,\) and \(7.\)

Therefore, \(R = \left\{ {\left( {2,8} \right),\left( {3,27} \right),\left( {5,125} \right),\left( {7,343} \right)} \right\}\)

Chapter 2 Ex.2.2 Question 8

Let \(A = \left\{ {x,y,z} \right\}\) and \(B = \left\{ {1,2} \right\}\). Find the number of relations from \(A\) to \(B.\)

Solution

It is given that \(A = \left\{ {x,y,z} \right\}\) and \(B = \left\{ {1,2} \right\}\).

Therefore, \(A \times B = \left\{ {\left( {x,1} \right),\left( {x,2} \right),\left( {y,1} \right),\left( {y,2} \right),\left( {z,1} \right),\left( {z,2} \right)} \right\}\)

Since \(n\left( {A \times B} \right) = 6\), the number of subsets of \(\left( {A \times B} \right) = {2^6}\).

Hence, the number of relations from \(A\) to \(B\) is \({2^6}\).

Chapter 2 Ex.2.2 Question 9

Let \(R\) be the relation on \(\bf{Z}\) defined by \(R = \left\{ {\left( {a,b} \right):a,b \in {\bf{Z}},a - b {\text{ is an integer}}} \right\}\). Find the domain and range of \(R.\)

Solution

It is given that \(R = \left\{ {\left( {a,b} \right):a,b \in {\bf{Z}},a - b {\text{ is an integer}}} \right\}\)

It is known that the difference between any two integers is always an integer.

Therefore,

Domain of \(R = {\bf{Z}}\)

Range of \(R = {\bf{Z}}\)