Functions

• Definition of function
• For $a \in A$, we write $f(a)$, the image of $a$ to denote the unique element of $B$ assigned to $a$
• Definition of image, preimage, and range

Injection and Surjection

• Definition of injection
• Definition of surjection
• Definition of bijective
  • Bijective functions also have inverses

Composition

• Definition of composition

Graphs

• Definition of graph
  • The graph is always a subset of the cartesian product

Floor and Ceiling

• $\lfloor x \rfloor = \max\{n \in \mathbb{Z} \ | \ n \le x\}$
• $\lceil x \rceil = \min\{n \in \mathbb{Z} \ | \ n \gt x\}$
• $\forall \ n \in \mathbb{Z}$, $\lceil n \rceil = \lfloor n \rfloor = n$

Properties

• $\lfloor x \rfloor = n \iff x \in [n, n+1) \lor n \in (x-1, x]$
• $\lceil x \rceil = n \iff x \in (n-1, n] \lor n \in [x, x+1)$
• $x - 1 \lt \lfloor x \rfloor \le x \le \lceil x \rceil \lt x + 1$
• $\lfloor -x \rfloor = -(x+1) = -\lceil x \rceil$
• $\lceil -x \rceil = -(x-1) = -\lfloor x \rfloor$
• $\lfloor x + n \rfloor = \lfloor x \rfloor + n$
• $\lceil x + n \rceil = \lceil x \rceil + n$

References

• Course slides §11.1: Functions
• Course slides §12.1: Function Composition, Floor and Ceiling Functions