Concept of Relations and Functions

1 Mark Questions

1. If <math>R = \{(a, a^3) : a \text{ is a prime number less than } \}</math>
2. be a relation. Find the range of R. Foreign 2014 Given, <math>R = \{(a, a^3): a \text{ is a prime number less}\}</math> than 5} We know that, 2 and 3 are the prime numbers less than 5. ∴ a can take values 2 and 3.

Then, <math>R = \{(2,2^3), (3,3^3)\} = \{(2,8), (3,27)\}</math> (1) Hence, the range of R is <math>\{8,27\}</math>. 2. If <math>f: \{1,3,4\} \rightarrow \{1,2,5\}</math> and <math>g: \{1,2,5\} \rightarrow \{1,3\}</math> given by <math>f = \{(1,2), (3,5), (4,1)\}</math> and <math>g = \{(1,3), (2,3), (5,1)\}</math>. Write down gof. All India 2014C

The functions <math>f:\{1,3,4\} \rightarrow \{1,2,5\}</math> and <math>g:\{1,2,5\} \rightarrow \{1,3\}</math> are defined as <math>f = \{(1,2), (3,5), (4,1)\}</math> and <math>g = \{(1,3), (2,3), (5,1)\}</math> :: <math>gof(1) = g(f(1)) = g(2) = 3</math> <math>[:: f(1) = 2 \text{ and } g(2) = 3]</math>

<math>gof(3) = g(f(3)) = g(5) = 1</math>

[:: <math>f(3) = 5</math> and <math>g(5) = 1</math>]

<math>gof(4) = g(f(4)) = g(1) = 3</math>

[:: <math>f(4) = 1</math> and <math>g(1) = 3</math>]

<math>gof = \{(1,3), (3,1), (4,3)\}</math> (1) ...

3. Let R is the equivalence relation in the set <math>A = \{0,1,2,3,4,5\}</math> given by <math>R = \{(a,b): 2 \text{ divides }\}</math>

1. b)}. Write the equivalence class [0]. Delhi 2014C

Given, <math>R = \{(a, b): 2 \text{ divides } (a-b)\}</math> Here, all even integers are related to zero, i.e. (0, 2)(0, 4). Hence, equivalence class of <math>[0] = \{2,4\}</math> (1)

4. If <math>R = \{(x, y) : x + 2y = 8\}</math> is a relation on N, then write the range of R. All India 2014

Given, the relation R is defined on the set of natural numbers, i.e. N as

<math display="block">R = \{(x, y) : x + 2y = 8\}</math>

To find the range of R, <math>x + 2y = 8</math> can be rewritten as <math>y = \frac{8-x}{2}</math>.

On putting <math>x = 2</math>, we get <math>y = \frac{8-2}{2} = 3</math>

On putting <math>x = 4</math>, we get <math>y = \frac{8-4}{2} = 2</math>

On putting <math>x = 6</math>, we get <math>y = \frac{8-6}{2} = 1</math>

As, <math>x, y \in N</math>, then <math>R = \{(2, 3), (4, 2), (6, 1)\}</math> Hence, range of relation is <math>\{3, 2, 1\}</math>. (1) 5. If <math>A = \{1, 2, 3\}</math>, <math>B = \{4, 5, 6, 7\}</math> and <math>f = \{(1, 4), (2, 5), (3, 6)\}</math> is a function from A to

1. State whether f is one-one or not. All India 2011

Given, <math>A = \{1, 2, 3\}</math> and <math>B = \{4, 5, 6, 7\}</math>

Now, <math>f: A \rightarrow B</math> is defined as

<math>f = \{(1, 4), (2, 5), (3, 6)\}</math>

Therefore, <math>f(1) = 4</math>, <math>f(2) = 5</math> and <math>f(3) = 6</math>.

It is seen that the images of distinct elements of A under f are distinct. So, f is one-one. (1)