11th NCERT Relatins and functions. Exercise 2.2 Questions 9

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Question (1)

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.Solution

Let A = {1, 2, 3, … , 14}.R : A → A

R = {(x, y): 3x – y = 0, where x, y ∈ A}.

R = { (1,3),(2,6),(3,9),(4,12)}

Domain of R = { 1, 2, 3, 4}

Codomain of R = { 1, 2, 3, ....14} = A.

Range of R = { 3, 6, 9, 12}

Question (2)

Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.Solution

R : N → N,R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}.

x = 1, 2, 3,

R = {(1,6),(2,7),(3,8)}

The domain of R = {1, 2, 3}

The Range of R = { 6, 7, 8}

Question (3)

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

A = {1, 2, 3, 5} and B = {4, 6, 9}.R : A → B

R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}

R = {(1,4), (1,6),(2,9), (3,4), (3,6),(5,4),(5,6)}

Question (4)

The given figure shows a relationship between the sets P and Q. write this relation(i) in set-builder form (ii) in roster form.

(iii) What is its domain and range?

Solution

R : P → Q(i) In set builder form.

R = { (x,y) : y = x - 2 , x ∈ P, y ∈ Q}

(ii) Roster form

R = {(5,3), (6,4), (7,5)}

(iii) Domain of R = { 5, 6, 7}

Range of R = { 3, 4, 5}

Question (5)

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by{(a, b): a, b ∈ A, b is exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Solution

A = {1, 2, 3, 4, 6}.R : A → A

R = {(a, b): a, b ∈ A, b is exactly divisible by a}.

(i) Roster form

R = { ( 1,1),(1,2), (1,3), (1,4), (1,6), (2,2), (2,4), (2,6), (3,3) ,(3,6), ( 4,4), (6,6)}

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

(iii) Range og R = { 1, 2, 3, 4, 6}

Question (6)

Determine the domain and range of the relation R defined by R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.Solution

R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.R = {(0,5), (1,6),(2,7), (3,8), (4,9), (5,10)}

So domain of r = { 0, 1, 2, 3, 4, 5}

Range of R = {5, 6, 7, 8, 9, 10}

Question (7)

Write the relation R = {(x, x3): x is a prime number less than 10} in roster form.Solution

R = {(x, x3): x is a prime number less than 10}So the relation R in roaster form is

R = { (2,8), (3,27), (5,125), (7,343)}

Question (8)

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.Solution

Let A = {x, y, z} and B = {1, 2}.n(a) = m = 3, n(B) = n = 2.

The number of relations = (2

= (2

= 2

= 64.

Question (9)

Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.Solution

R : Z → ZR = {(a, b): a, b ∈ Z, a – b is an integer}. R = {......., ( -1,0),(-1,1), (-1,2),....,(0,-1),(0,0).....}

So domain of R = {....-2, -1, 0, 1, 2,....} = Z

And range of R = { ....-2, -1, 0, 1, 2,....} = Z