1. Functions#

What is a function?#

Define 2 sets of elements, we define function as a mapping of elements from 1 set into another set.

\[ \mathcal{f}:X \rightarrow Y \]

We say this as \(f\) maps \(X\) into \(Y\). \(X\) is called the domain of f, and Y is called the co-domain of f. The set \(f(X)\) is called the range of f.

We can define mathematically the range of \(f\),i.e. \(f(X)\) as:

\[ f(X) = \{f(x) | x \in X\} \]

Couple of things to remember:

  • In general, the range of \(f\) does not equal to the co-domain of \(f\). That’s why we say it as \(f\) maps \(X\) into \(Y\).

  • However, by defition, \(f\) will be called a function if for all elements \(x \in X\), there exists a unique value \(f(x)\),i.e. the function assigns a unique value \(f(x) \forall x \in X\).

Properties of functions#

There are 3 properties: injective, surjective and bijective.

Injective: A function is called injective (“one-to-one”) iff

\[f(x_1) = f(x_2) \leftrightarrow x_1 = x_2\]

This means that an element in \(Y\) cannot be a mapping of more than 1 element in \(X\). Equivalently, \(x_1 \neq x_2 \leftrightarrow f(x_1) \neq f(x_2)\).

Surjective: A function is called surjective (“onto”) iff

\[\forall y \in Y, \exists x \in X \ni y = f(x)\]

This means that every element in \(Y\) is a result of the mapping from \(X\) through \(f\). We can think of this as the range of \(f\) expands the entire co-domain.

Bijective: A function is called bijective if it is both surjective and injective.

\[\forall y \in Y, \exists! x \in X \ni y = f(x)\]

Injectivity comes in at the \(\exists!\),i.e. a unique mapping.