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, co-domain and range.
It is given that, R:A→A and R={(x, y) :3x – y = 0, where x, y∈A}
i.e. R=(x, y) :3x=y, where x, y∈A}
∴ R={(1,3),(2,6),(3,9),(4,12)}
Domain = {1,2,3,4}
Range = {3,6,9,12}
Co-domain = Set A = {1,2,3,...,14}

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.
R = {(x, y) :y =x+5, x is a natural number less than 4; x, y ∈N}
x is a natural number less than 4 so values of x can be 1,2,3
∴ R={(1,6),(2,7),(3,8)}
Domain = {1,2,3}
Range = {6,7,8}

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.
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)}

4. The Fig 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?

According to the Fig., P={5,6,7} and Q={3,4,5}
(i) R={(x,y): y=x-2, x∈P, y∈Q}
(ii) R={(5,3),(6,4),(7,5)}
Domain = {5,6,7}
Range = {3,4,5}

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
R= {(a,b): a,b ∈A, b is exactly divisible by a}
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) Find the domain of R
Domain = {1,2,3,4,6}

(iii) Find the range of R.
Range= {1,2,3,4,6}

6. Determine the domain and range of the relation R defined by R={(x, x+5) : x∈{0, 1, 2, 3, 4, 5}}.
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= {0,1,2,3,4,5}
Range= {5,6,7,8,9,10}

7. Write the relation R = {(x,$\displaystyle \small x^{3}$) : x is a prime number less than 10} in roster form.
R = {(x,$\displaystyle \small x^{3}$) : x is a prime number less than 10}
Prime numbers less than 10 are 2,3,5,7
R= {(2,8),(3,27),(5,125),(7,343)}

8. Let A ={x, y, z} and B ={1, 2}. Find the number of relations from A to B.
A ={x, y, z}, B ={1, 2}
n(A)=p=3, n(B)=q=2
n(A⨯B)= pq = 6
Number of relations from A to B=$\displaystyle \small 2^{pq}$=$\displaystyle \small 2^{6}$=64

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.
R = {(a,b ): a, b ∈ Z, a–b is an integer}
We know that difference between any two integers is always an integer.
∴ Domain = Z
Range = Z