Injectivity, Surjectivity, Bijectivity

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

Example 4. Consider the function $f:\mathbb{N}\to \{1,4,9,16,\dots \}$ given by $f(n)=n^2$. This is a bijective function. The inverse map $f^{-1}$ is given by: $$\begin{array}{rl}{f}^{-1}:\{1,4,9,16,25,\dots \}& \to \mathbb{N}\\ m& \mapsto \sqrt{m}\\ \text{or: }{n}^2& \mapsto n\end{array}$$

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

Which of the functions are injective, surjective or bijective?