# Function: Notations and Rules

## What is a Function?

A function is a relation such that no two distinct members have the same first coordinate in its graph. $f$ is a function iff

1. The members of $f$ are ordered pairs, i.e., $xfy \Rightarrow (x,y) \in f$ .
2. If ordered pairs $(x, y)$ and $(x, z)$ are members of $f$ , i.e., $(x,y) \in f$ and $(x,z) \in f$ , then $y=z$

## Notation for functions

A function is usually defined as ordered-pairs, and $\text{ordered pair } (x,y) \in \text{function } f$ so that $xfy$ is (was) a way to represent where $x$ is an argument of $f$ and $y$ is image (value) of $f$ .

The set of arguments is called the domain of the function, while the set of images is termed as the range of the function.

Other popular notations for $(x,y)\in f$ and $xfy$ are: $y : xf$ , $y=f(x)$ , $y=fx$ , $y=x^f$ .

## Into Function

A function $f$ is into $Y$ iff the range of $f$ is a subset of $Y$ . i.e., $R_f \subset Y$

## Onto Function

A function $f$ is onto $Y$ iff the range of $f$ is $Y$ . i.e., $R_f=Y$

Generally, a mapping is represented by $f : X \rightarrow Y$ .

## One-to-One function

A function is called one-to-one if it maps distinct elements onto distinct elements.
A function $f$ is one-to-one iff $x_1 \ne x_2 \Leftrightarrow f(x_1) \ne f(x_2)$ and $x_1 = x_2 \Leftrightarrow f(x_1)=f(x_2)$

## Restriction of Function

If $f : X \rightarrow Y$ and if $A \subseteq X$ , then $f \cap (A \times Y)$ is a function on $A \ \text{into } \ Y$ , called the restriction of $f$ to $A$ and $f \cap (A \times Y)$ is usually abbrevated by $f|A$ .

## Extension of function

The function $f$ is an extension of a function $g$ iff $g \subseteq f$ .