Relations & Functions — Formula & Trap Sheet

Every notation, formula, and trap across all 5 modules, on one page. Scan this the night before — if something here doesn't click, go back to that module's mastery check.

⚠️ The #1 trap in this chapter

The greatest integer function $[x]$ ALWAYS rounds down, even for negatives: $[-2.3] = -3$, not $-2$. Students almost always round toward zero out of habit — that instinct is wrong here.

Module 1

Cartesian Products of Sets

$A \times B$ $(a,b)$ $n(A\times B)=n(A)n(B)$ $A\times A\times A$

If $n(A)=p$, $n(B)=q$, then $n(A\times B) = pq$ — multiply the counts

$A\times B \ne B\times A$ in general — order matters, just like inside the ordered pair.
If either A or B is $\varnothing$, then $A\times B=\varnothing$ too — nothing to pair with nothing.

Module 2

Relations

$R \subset A\times B$ Domain Range Codomain

Total relations from A to B $= 2^{pq}$ where $n(A)=p, n(B)=q$

Range $\subseteq$ Codomain always — only equal when every element of B is somebody's image.
An element of A with NO arrow is simply not in the domain — allowed for relations, but not allowed for functions (next module).

Module 3

Functions

$f:A\to B$ $f(a)=b$ image preimage

Function $\iff$ every element of A has EXACTLY one image — not zero, not two or more

Repeated outputs are fine — only repeated inputs with different outputs break a function.
$f(a)=b$: $a$ is the preimage, $b$ is the image — don't flip the direction.

Module 4

Seven Functions Worth Memorizing

identity constant modulus signum greatest integer

Signum range $=\{-1,0,1\}$ · Modulus range $=[0,\infty)$ · Greatest integer range $=\mathbb{Z}$

$[x]$ ALWAYS rounds down, even for negatives — $[-2.3]=-3$, the #1 trap in this topic.
Domain of $\frac{1}{x}$-style rational functions: exclude wherever the denominator is zero, state it as $\mathbb{R}-\{\text{those points}\}$.

Module 5

Algebra of Real Functions

$(f+g)(x)$ $(f-g)(x)$ $(\alpha f)(x)$ $(fg)(x)$ $(f/g)(x)$

All five operations are done POINTWISE — compute $f(x)$ and $g(x)$ separately, then combine

Scalar multiple $\alpha f$ (one function times a number) and product $fg$ (two functions multiplied) look similar but are completely different operations.
$f/g$ always needs a domain restriction: wherever $g(x)=0$ gets excluded, even if the original domain didn't.