5-6. Properties of functions

Properties of functions

Injective function

Injective function

A function $f:A\to B$ is called a one-to-one or injective function if it maps distinct elements of $A$ to distinct elements of $B$.

Example of a Injective mapping

Alternatively: $f$ is one-to-one if for all elements $x,y$ in $A$, if $x\ne y$ then $f(x)\ne f(y)$.

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Surjective function

Surjective function

A function $f:A\to B$ is called onto or a surjective function if every element $b$ in $B$ can be obtained as $b = f(a)$ for some $a$ in $A$. To summarise, the entire codomain must be mapped to the domain.

Example of a Surjective mapping

In general, this might not be the case. If $f$ is an $A\to B$ function, we know that for every $a\in A$, we have $f(a)\in B$. If the range of $f$ can be a proper subset of $B$ (which is the codomain), then it is not onto.

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Bijection

Bijection (function)

A function is called a bijection if it is both an [[Injective function|injective]] and [[Surjective function|surjective]] function.

Example of a Bijection mapping	Example of neither injective or surjective mapping

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Inverse Functions

Inverse (function)

Bijections always come in pairs, if $f:A\to B$ is a bijection, then there is a function $\boxed{f^{-1}:B\to A}$, it is called the inverse of $f$, defined by:

$$f^{-1}(b)=a\text{ whenever }f(a)=b$$

This also means $f^{-1}$ is also a bijection, so we have:

$$\begin{array}{rl}{f}^{-1}(f(a))& =a\text{ for every }a\in A\\ f^{-1}(f(b))& =b\text{ for every }b\in B\end{array}$$

For example, let Odd and Even be the sets of odd and even natural numbers, respectively. Define a function $f:\text{Odd }\to \text{ Even}$ by $f(n)=n-1$. Then $f$ is a bijection and its inverse $f^{-1}:\text{Even }\to \text{ Odd}$ is defined as $f^{-1}(n)=n+1$.

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Identity Function

Identity function

For every set $A$, its identity function ${\text{id}}_A:A\to A$ is defined for all $a\in A$ as: $$ \text{id}_A(a) = a $$

${\text{id}}_A$ is also a bijection and its inverse is itself.

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Characteristic function

Characteristic function

Let $S$ be a set. For every subset $A\subseteq S$, its characteristic function, $f_A:S\to \{0,1\}$ is defined for all $x\in S$ by: $$ f_A(x) = \begin{cases} 1, &\text{if } x \in A, \\ 0, &\text{if } x \notin A \end{cases} $$

If $A\subseteq \mathbb{N}$ then $f_a:\mathbb{N}\to \{0,1\}$ can be represented by an infinite 0-1 sequence.

For example, given $\text{Odd }\subseteq \mathbb{N}$, $f_{\text{odd}}=0101010101[..]$.

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