Functions
7.0 Types of Functions
7.0 Types of Functions
1. ONE-ONE Function (Injection):
A function $f:A \to B$ is said to be a one-one function or an injection if different elements of $A$ have different images in $B$. $$\begin{equation} \begin{aligned} f:A \to B\;\;is\;\;one - one \\ \Leftrightarrow \;a \ne b \Rightarrow f(a) \ne f(b)\;\;for\;\;all\;a,b \in A \\ \Leftrightarrow f(a) = f(b) \Rightarrow a = b\;\;for\;\;all\;a,b \in A \\\end{aligned} \end{equation} $$
2. MANY-ONE Function:
A function $f:A \to B$ is said to be a many-one function if two or more elements of $A$ have the same image in $B$.
Thus, $f:A \to B$ is a many one function if there exists $x,\;y \in A$ such that $x \ne y$ but $f(x) = f(y)$.
3. ONTO Function (Surjection):
A function $f:A \to B$ is said to be an onto function or a surjection if every element of $B$ is the f-image of some element of $A$.
In other words, $f(A) = B$ or the range and co-domain of the function $f$ is one and the same.
Thus, $f:A \to B$ is a surjection iff for each $b \in B$, there exists $a \in A$ such that $f(a) = b$.
4. INTO Function:
A function $f:A \to B$ is said to be an into function if there exists an element in $B$ having no pre-image in $A$.
In other words, $f(A) = B$ is an into function if it is not an onto function.
5. Bijection or ONE-ONE ONTO Function:
A function $f:A \to B$ is a bijection if it is one-one as well as onto.
i.e., a function, $f:A \to B$ is a bijection, if it is $$\begin{equation} \begin{aligned} i.\;one - one,\;i.e.\;f(x) = f(y) \Rightarrow x = y\;for\;all\;\;x,y \in A \\ ii.\;onto\;i.e.\;for\;all\;y \in B,\;there\;exists\;x \in A\;such\;that\;f(x) = y \\\end{aligned} \end{equation} $$
Note: From above definitions, if $A$ and $B$ are two finite sets and $f:A \to B$ is a function, then, $$\begin{equation} \begin{aligned} i.\;f\;is\;an\;injection\; \Rightarrow \;n(A) \le \;n(B) \\ ii.\;f\;is\;a\;surjection\; \Rightarrow \;n(B) \le \;n(A) \\ iii.\;f\;is\;a\;bijection\; \Rightarrow \;n(A) = n(B) \\\end{aligned} \end{equation} $$
