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 abbreviated by $ f|A$ .

Extension of function

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