These are important notions for maps.

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**Definition 1** (Injective, surjective and
bijective). *A map \(f: X \to Y\) is
called*

*injective* if every fiber of \(f\) has only one element: \(x_1 \neq x_2 \Rightarrow f(x_1) \neq
f(x_2)\).

*surjective* if \(\mathrm{Ran}(f)=Y\). With quantifiers:
\(\forall y\in Y~ \exists x\in X \,:\,
f(x)=y\).

*bijective* if \(f\)
is both injective and surjective.

**Example 2**. Define the function that maps each
student to her or his chair. This means that \(X\) is the set of all students in the room,
and \(Y\) the set of all chairs in the
room.

well-defined: every student has a chair

surjective: every chair is taken

injective: on each chair there is no more than one
student

bijective: every student has his/her own chair, and no chair is
empty

**Rule of Thumb 3**. Surjective,
injective, bijective A map \(f: X \rightarrow Y\) is \[\begin{aligned}
\text{surjective}\ &\Leftrightarrow\ \text{at each } y\in Y \text{
arrives at least one arrow} \\
&\Leftrightarrow\ f(X)=Y\\
&\Leftrightarrow\ \text{the equation } f(x)=y \text{ has for all }
y\in Y \text{ a solution} \\
\\
\text{injective}\ &\Leftrightarrow\ \text{at each } y\in Y \text{
arrives at most one arrow}\\
&\Leftrightarrow\ \left( x_1 \neq x_2\quad \Rightarrow\quad
f(x_1)\neq f(x_2) \right) \\
&\Leftrightarrow\ \left( f(x_1)=f(x_2)\quad \Rightarrow\quad x_1=x_2
\right) \\
&\Leftrightarrow\ \text{the equation } f(x)=y \text{ has for all }
y\in f(X) \text{ a unique solution} \\
\\
\text{bijective}\ &\Leftrightarrow\ \text{at each } y\in Y \text{
arrives exactly one arrow} \\
&\Leftrightarrow\ \text{the equation } f(x)=y \text{ has for all }
y\in Y \text{ a unique solution}
\end{aligned}\]

Thus, if \(f\) is bijective, there
is a well defined inverse map \[\begin{aligned}
f^{-1}:Y&\to X\\
y &\mapsto x \text{ where } f(x)=y \,.
\end{aligned}\] Then \(f\) is
called *invertible* and \(f^{-1}\) is called the *inverse map*
of \(f\).

**Example 4**. Consider the function \(f: \mathbb{N} \rightarrow \{1, 4, 9, 16,
\ldots\}\) given by \(f(n) =
n^2\). This is a bijective function. The inverse map \(f^{-1}\) is given by: \[\begin{aligned}
f^{-1}:\lbrace1,4,9,16,25,\dots \rbrace &\rightarrow \mathbb{N}
\\
m & \mapsto \sqrt{m} \\
\text{or: } n^2 &\mapsto n
\end{aligned}\]

**Example 5**. For a function \(f:\mathbb{R}\rightarrow\mathbb{R}\), we can
sketch the graph \(\lbrace(x,f(x)): x\in
X\rbrace\) in the \(x\)-\(y\)-plane:

Which of the functions are injective, surjective or bijective?

These notions might seem a little bit off-putting, but we will use
them so often that you need to get use to them. Maybe the video will
help you as well.

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