### Relations and Functions Exercise 2

## Question:

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

## Answer:

Domain: {0,1,2,3,4,5} Range: {5,6,7,8,9,10}

## Question:

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

## Answer:

Answer: The number of relations from A to B is 8.

Explanation: A relation from A to B is a set of ordered pairs (x,y) where x is an element of A and y is an element of B.

There are 3 elements in A (x,y,z) and 2 elements in B (1,2).

Therefore, there are 3 possible elements of A that can be paired with each element of B.

This means that there are 3x2 = 6 possible ordered pairs.

However, since each ordered pair can be reversed (e.g. (x,2) can also be written as (2,x)), the total number of relations from A to B is doubled, giving us 8 relations.

## Question:

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:

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

## Question:

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.

## Answer:

Domain: A = {1,2,3,….,14}

Co-domain: A = {1,2,3,….,14}

Range: {3, 6, 9, 12, 15}

## Question:

Let A={1,2,3,4,6} and 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:

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

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

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

## Question:

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

## Answer:

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

## Question:

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:

Domain: Z Range: Z

## Question:

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 (i) roster form (ii) an arrow diagram. Write down the domain and range or R.

## Answer:

Roster Form: {(0,5), (1,6), (2,7), (3,8)}

Arrow Diagram:

0 → 5 1 → 6 2 → 7 3 → 8

Domain of R: {0, 1, 2, 3} Range of R: {5, 6, 7, 8}