**Exercise 2.2 Relations and Functions**

**Question and Answers**

**Class 11 – Maths**

Class | Class 11 |

Subject | Mathematics |

Chapter Name | Relations and Functions |

Chapter No. | Chapter 2 |

Exercise | Exercise 2.2 |

Category | Class 11 Maths NCERT Solutions |

**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.**

**Answer** The relation R from A to A is given as:

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

= {(*x*, *y*): 3*x* = *y*, where *x*, *y* ∈ A}

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

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

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

The whole set A is the codomain of the relation R.

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

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

Hence, 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.**

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

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

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

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

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

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

Hence, 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.**

**Answer** A = {1, 2, 3, 5} and B = {4, 6, 9}

The relation from A to B is given as

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 Fig 2.7 shows a relationship between the sets P and Q. Write this relation**

** (i) in set-builder form **

**(ii) roster form. **

**What is its domain and range?**

**Answer** P = {5, 6, 7}, Q = {3, 4, 5}

The relation between P and Q:

(i) Set-builder form

R = {(*x, y*): *y = x* – 2; *x* ∈ P} or R = {(*x, y*): *y = x* – 2 for *x* = 5, 6, 7}

(ii) Roster form

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

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.**

**Answer** A = {1, 2, 3, 4, 6} and relation R = {(*a*, *b*): *a*, *b* ∈ A, *b* is exactly divisible by *a*}

(i) 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 of 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}}.**

**Answer **Relation R = {(*x*, *x* + 5): *x* ∈ {0, 1, 2, 3, 4, 5}}

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

Domain of R = {0, 1, 2, 3, 4, 5} and,

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

**Question 7 Write the relation R = {(x, x ^{3} ) : x is a prime number less than 10} in roster form.**

**Answer** Relation R = {(*x*, *x*^{3}): *x *is a prime number less than 10}

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

∴ 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.**

**Answer** A = {*x*, *y*, z} and B = {1, 2}

A × B = {(*x*, 1), (*x*, 2), (*y*, 1), (*y*, 2), (*z*, 1), (*z*, 2)}

As *n*(A × B) = 6, the number of subsets of A × B will be 2^{6}.

Thus, the number of relations from A to B is 2^{6}.

**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.**

**Answer** Relation R = {(*a*, *b*): *a*, *b* ∈ Z, *a *– *b* is an integer}

We know that the difference between any two integers is always an integer.

∴ Domain of R = Z and Range of R = Z

## Leave a Reply