### Exercise 2.2

1. Let \[{\rm{A = }}\left\{ {{\rm{1,2,3}}...{\rm{14}}} \right\}\]. Define a relation \[{\rm{R}}\] from \[{\rm{A}}\] to \[{\rm{A}}\] by \[{\rm{R = }}\ left\{ {\left( {{\rm{x,y}}} \right){\rm{:3x - y = 0}}} \right\}\], where \[{\rm{x, y}} \in {\rm{A}}\]. Write your domain, reach and image.

Answer:The ratio \[{\rm{R}}\] of \[{\rm{A}}\] for \[{\rm{A}}\] is given as \[{\rm{R = }} \left\{ {\left( {{\rm{x,y}}} \right){\rm{:3x - y = 0}}} \right\}\] at \[{\rm{x, y}} \in {\rm{A}}\]

oder seja, \[{\rm{R = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{:3x = y}}} \right\} \ ] wobei \[{\rm{x,y}} \in {\rm{A}}\]

\[\so {\rm{R = }}\left\{ {\left( {{\rm{1,3}}} \right){\rm{,}}\left( {{\rm{2 ,6}}} \right){\rm{,}}\left( {{\rm{3,9}}} \right){\rm{,}}\left( {{\rm{4,12 }}} \gut gut\}\]

The domain of \[{\rm{R}}\] is the set of all first elements of ordered pairs in the relation.

Also Definitionsbereich von \[{\rm{R = }}\left\{{{\rm{1,2,3,4}}} \right\}\].

The complete set \[{\rm{A}}\] is the domain of the relation \[{\rm{R}}\].

Hence condom of \[{\rm{R = }}\left\{ {{\rm{1,2,3}}...{\rm{14}}} \right\}\]

The domain of \[{\rm{R}}\] is the set of all second elements of the ordered pairs in the relation.

Therefore range of \[{\rm{R = }}\left\{{{\rm{3,6,9,12}}} \right\}\]

2. Define a relation \[{\rm{R}}\]over the set \[{\rm{N}}\] โโโโโโโโโโโโโof natural numbers by \[{\ rm{ R = }}\left \{ { \ left( {{\rm{x,y}}} \right){\rm{:y = x + 5, \text{x is an integer less than 4 }; x,y}} \in {\rm{N}}} \right\}\]. Describe this relationship using the list form. Write the domain and image.

Answer:if you give us this

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

The natural numbers less than 4 are 1, 2, and 3.

Daher ist \[{\rm{R = }}\left\{ {\left( {{\rm{1,6}}} \right){\rm{,}}\left( {{\rm {2 ,7}}} \right){\rm{,}}\left( {{\rm{3,8}}} \right)} \right\}\]

The domain of \[{\rm{R}}\] is the set of all first elements of ordered pairs in the relation.

Also Definitionsbereich von \[{\rm{R = }}\left\{{{\rm{1,2,3}}} \right\}\]

The domain of \[{\rm{R}}\] is the set of all second elements of the ordered pairs in the relation.

Therefore range of \[{\rm{R = }}\left\{{{\rm{6,7,8}}} \right\}\]

3.\[{\rm{A = }}\left\{ {{\rm{1,2,3,5}}} \right\}\]y \[{\rm{B = }}\left \{ {{\rm{4,6,9}}} \right\}\]. Define a relation \[{\rm{R}}\]from \[{\rm{A}}\] to \[{\rm{B}}\] times \[{\rm{R = }}\ left\{ {\left( {{\rm{x,y}}} \right){\rm{\text{: the difference between x and y is odd; x}}} \in {\rm{A, y}} \in {\rm{B}}} \right\}\]. Enter \[{\rm{R}}\] in the list.

Answer:if you give us this

\[{\rm{A = }}\left\{ {{\rm{1,2,3,5}}} \right\}\] y \[{\rm{B = }}\left\{ {{\rm{4,6,9}}} \derecho\}\]

The relationship is given by

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

Daher ist \[{\rm{R = }}\left\{ {\left( {{\rm{1,4}}} \right){\rm{,}}\left( {{\rm {1 ,6}}} \right){\rm{,}}\left( {{\rm{2,9}}} \right){\rm{,}}\left( {{\rm{3 ,4 }}} \right){\rm{,}}\left( {{\rm{3,6}}} \right){\rm{,}}\left( {{\rm{5,4 }} } \right){\rm{,}}\left( {{\rm{5,6}}} \right)} \right\}\]

4. The given figure shows a relation between the sets \[{\rm{P}}\] and \[{\rm{Q}}\]. Write that relationship down

(i) In the form of a set constructor

(ii) In Listenform

What is your domain and image?

(image will be uploaded soon)

Answer:

(UE)According to the given diagram \[{\rm{P = }}\left\{ {{\rm{5,6,7}}} \right\}\]

mi \[{\rm{Q = }}\left\{ {{\rm{3,4,5}}} \right\}\]

Hence the relational form of the set constructor

\[{\rm{R = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{: y = x - 2; x}} \in {\rm{P}}} \right\}\] oder

\[{\rm{R = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{: y = x - 2 \text{for} x = 5 ,6,7}}} \direita\}\]

(ii)According to the given diagram \[{\rm{P = }}\left\{ {{\rm{5,6,7}}} \right\}\]

mi \[{\rm{Q = }}\left\{ {{\rm{3,4,5}}} \right\}\]

Hence the list form of relationships

\[{\rm{R = }}\left\{ {\left( {{\rm{5,3}}} \right){\rm{,}}\left( {{\rm{6,4 }}} \right){\rm{,}}\left( {{\rm{7.5}}} \right)} \right\}\]

5. Let \[{\rm{A = }}\left\{ {{\rm{1,2,3,4,6}}} \right\}\]. Let \[{\rm{R}}\] be the relation on \[{\rm{A}}\] defined by \[\left\{ {\left( {{\rm{a,b}}} \ right){\rm{: a,b}} \in {\rm{A, \text{ b is exactly divisible by a}}}} \right\}\].

(i) Write \[{\rm{R}}\] in list form

(ii) Find the domain of \[{\rm{R}}\]

(iii) Find the image of \[{\rm{R}}\]

Answer:

(UE)We know that \[{\rm{A = }}\left\{{{\rm{1,2,3,4,6}}} \right\}\]

e \[{\rm{R}} = \left\{ {\left( {{\rm{a,b}}} \right){\rm{: a,b}} \in {\rm{A , \text{b รฉ exatamente dividido por a}}}} \right\}\]

Therefore the list form of the relation is \[{\rm{R}}\]

\[{\rm{R = }}\left\{ {\left( {{\rm{1,1}}} \right){\rm{,}}\left( {{\rm{1,2 }}} \derecha){\rm{,}}\izquierda( {{\rm{1,3}}} \derecha){\rm{,}}\izquierda( {{\rm{1,4}} } \right){\rm{,}}\left( {{\rm{1,6}}} \right){\rm{,}}\left( {{\rm{2,2}}} \ derecha){\rm{,}}\left( {{\rm{2,4}}} \right){\rm{,}}\left( {{\rm{2,6}}} \right) {\rm{,}}\left( {{\rm{3,3}}} \right){\rm{,}}\left( {{\rm{3,6}}} \right){\ rm{,}}\left( {{\rm{4,4}}} \right){\rm{,}}\left( {{\rm{6,6}}} \right)} \right\ }\]

(ii)The domain of \[{\rm{R}}\] is \[\left\{ {{\rm{1,2,3,4,6}}} \right\}\]

(iii)The range of \[{\rm{R}}\] is \[\left\{ {{\rm{1,2,3,4,6}}} \right\}\]

6. Determine the domain and domain of the relation \[{\rm{R}}\] defined by \[{\rm{R = }}\left\{ {\left( {{\rm{x,x + 5 }}} \right){\rm{:x}} \in \left\{ {{\rm{0,1,2,3,4,5}}} \right\}} \right\}\ ]

Answer:We know that the relation \[{\rm{R}}\] is given by

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

Daher ist \[{\rm{R = }}\left\{ {\left( {{\rm{0.5}}} \right){\rm{,}}\left( {{\rm{1 ,6 }}} \right){\rm{,}}\left( {{\rm{2,7}}} \right){\rm{,}}\left( {{\rm{3,8 }} } \right){\rm{,}}\left( {{\rm{4,9}}} \right){\rm{,}}\left( {{\rm{5,10}} } \ gut gut\}\]

Dominio de \[{\rm{R = }}\left\{{{\rm{0,1,2,3,4,5}}} \right\}\]

Rango de \[{\rm{R = }}\left\{ {{\rm{5,6,7,8,9,10}}} \right\}\]

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

Answer:if you give us this \[{\rm{R = }}\left\{\left( {{\rm{x,}}{{\rm{x}}^{\rm{3}}}} \right){\ rm {\text{:x is a prime less than 10}}}} \right\}\]

The prime numbers less than 10 are 2, 3, 5, and 7.

Por lo tanto, \[R = \left\{ {\left( {{\rm{2.8}}} \right){\rm{,}}\left( {{\rm{3.27}}} \ right) {\ rm{,}}\left( {{\rm{5,125}}} \right){\rm{,}}\left( {{\rm{7,343}}} \right)} \right\ }\ ] es el formelio de lista.

8. Let \[{\rm{A = }}\left\{ {{\rm{x,y,z}}} \right\}\] and \[{\rm{B = }}\left\ { {{\rm{1,2}}} \right\}\]. Find the number of ratios from \[{\rm{A}}\] to \[{\rm{B}}\].

Answer:Angenommen, \[{\rm{A = }}\left\{ {{\rm{x,y,z}}} \right\}\] und \[{\rm{B = }}\left \{ {{\rm{1,2}}} \right\}\].

Daher ist \[{\rm{A \times B = }}\left\{ {\left( {{\rm{x,1}}} \right){\rm{,}}\left( {{\ rm{x,2}}} \right){\rm{,}}\left( {{\rm{y,1}}} \right){\rm{,}}\left( {{\rm{ y,2}}} \right){\rm{,}}\left( {{\rm{z,1}}} \right){\rm{,}}\left( {{\rm{z, 2}}} \direita)} \direita\}\]

Since \[{\rm{n(A \times B) = 6}}\], or number of subsets of \[{\rm{A \times B}}\] is \[{{\rm{2 } }^{\rm{6}}}\].

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

Answer:Tenemos \[{\rm{R = }}\left\{ {\left( {{\rm{a,b}}} \right){\rm{: a,b}} \in {\ rm{ Z , a - b \text{es un nรบmero entero}}}} \right\}\]

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

Hence the domain of \[{\rm{R = Z}}\]

And the range of \[{\rm{R = Z}}\]

### NCERT Solutions for Class Math 11 Chapter 2 Relations and Functions Exercise 2.2

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