Functions
1.0 Definitions
2.0 Relation
3.0 Types of Relation
4.0 Functions
5.0 Standard Real Functions and their Graphical Representation
5.10 Reciprocal Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
6.0 Operations on Real Functions
7.0 Types of Functions
8.0 Composition of a Function
9.0 Inverse of a Function
2.2 Representation of a Relation
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
A relation from $A$ to $B$ can be represented in any of these ways:
- Roster
- Set-Builder
- Arrow Diagram
- Lattice
These methods are explained with the help of examples.
Example 11. Write in the given relations in roster form
| i. A relation $R$ from $A = \{ 2,3,4,5,6\} $ to $B = \{ 1,2,3\} $ defined by $x = y + 1,\;x \in A\;and\;y \in B$ | iii. A relation $R$ on the set $\{ 1,2,3,4,5\} $ defined by $(x,y) \in R \Leftrightarrow x\;is\;relatively\;prime\;to\;y$ |
| ii. A relation from $A = \{ 3,7\} $ to $B = \{ 5,9\} $ defined by $R = \{ (a,b):a \in A\;and\;b \in B,\;a - b\;is\;odd\} $ | iv. A relation $R$ on the set of $N$ of natural numbers defined by $R = \{ (a,b):\;b = a + 5,\;\;a \le 4,\;a,b \in N\} $ |
Solution:
| Relation Defined | Relation in Roster Form |
| i. A relation $R$ from $A = \{ 2,3,4,5,6\} $ to $B = \{ 1,2,3\} $ defined by $x = y + 1,\;x \in A\;and\;y \in B$ | The possible ordered pairs are, $(2,1),(3,2),(4,3)$. Thus, the relation in roster form is $R = \{ (2,1),(3,2),(4,3)\} $ |
| ii. A relation from $A = \{ 3,7\} $ to $B = \{ 5,9\} $ defined by $R = \{ (a,b):a \in A\;and\;b \in B,\;a - b\;is\;odd\} $ | It is known that Odd - Odd = Even. Since both sets contain only odd numbers, there is no possible ordered pair that obeys the condition. Thus, the relation in roster form is $R = \{ \} $ |
| iii. A relation $R$ on the set $\{ 1,2,3,4,5,6\} $ defined by $(x,y) \in R \Leftrightarrow x\;is\;relatively\;prime\;to\;y$ | Since $x$ is relatively prime to $y$, the possible ordered pairs are, $(2,3),(2,5),(3,2),(3,4),(3,5),(5,2),(5,3),(5,4),(5,6)$ Thus, the relation in roster form is $R = \{ (2,3),(2,5),(3,2),(3,4),(3,5),(5,2),(5,3),(5,4),(5,6)\} $ |
| iv. A relation $R$ on the set of $N$ of natural numbers defined by $R = \{ (a,b):\;b = a + 5,\;\;a \le 4,\;a,b \in N\} $ | The possible ordered pairs are, $(1,6),(2,7),(3,8),(4,9)$. Thus, the relation in roster form is $R = \{ (1,6),(2,7),(3,8),(4,9)\} $ |
