**Question** 1:

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(*x*, *y*): 3*x* – *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}

i.e., R = {(*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.

∴Domain of R = {1, 2, 3, 4}

The whole set A is the codomainof the relation R.

∴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.

∴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.

∴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.

∴ Domain of R = {1, 2, 3}

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

∴ 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}

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.

What is its domain and range?

**Answer**:

According to the given figure, P = {5, 6, 7}, Q = {3, 4, 5}

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

(ii) 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}, 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**:

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}

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**:

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**:

It is given that A = {*x*, *y*, z} and B = {1, 2}.

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

Since *n*(A × B) = 6, the number of subsets of A × B is 2^{6}.

Therefore, 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**:

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

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

∴Domain of R = **Z**

Range of R = **Z**