Sets — Formula & Trap Sheet

Every notation, formula, and trap across all 6 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

$\varnothing$ has 0 elements. $\{0\}$ has 1 element (the number zero). $\{\varnothing\}$ also has 1 element (the empty set, sitting inside another set). All three are different objects — this single confusion costs more marks in this chapter than anything else.

Module 1

What is a Set?

$a \in A$ $b \notin A$ roster form set-builder form

Roster: $\{2,4,6\}$ · Set-builder: $\{x : x \text{ is even}, x<7\}$

A description using a subjective word ("best", "most talented", "renowned") is NOT a set.
Order and repetition don't matter in roster form: $\{1,2,3\} = \{3,2,1\} = \{2,2,1,3,3\}$.

Module 2

Types of Sets

$\varnothing$ $n(S)$ finite / infinite equal sets

$\varnothing \ne \{0\} \ne \{\varnothing\}$ — zero, one, and one elements respectively

$\varnothing$ is considered FINITE — it has a definite count (zero).
To check $A=B$: verify BOTH directions — every element of A in B, and every element of B in A.

Module 3

Subsets & Intervals

$A \subset B$ $A \not\subset B$ $(a,b)$ open $[a,b]$ closed

n elements $\Rightarrow$ exactly $2^n$ subsets

$\varnothing \subset A$ is ALWAYS true — write it as your first line in any subset question.
$a \in A$ (element) vs. $\{a\} \subset A$ (subset) — mixing these two up is the single biggest mark-loser in this topic.
$[a,b)$ excludes $b$; $(a,b]$ excludes $a$ — the bracket shape tells you which side is closed.

Module 4

Venn Diagrams & Universal Set

$U$ rectangle = U circle = subset

U is drawn as a rectangle; every subset is a circle inside it — never the reverse

There's no single "correct" $U$ — if asked to pick one, choose the smallest set containing everything discussed.
Elements outside every circle still belong inside the rectangle (inside $U$, outside all subsets).

Module 5

Union, Intersection & Difference

$A \cup B$ $A \cap B$ $A - B$ disjoint: $A\cap B=\varnothing$

$A \cap (B \cup C) = (A\cap B) \cup (A \cap C)$ — distributive law

$A-B \ne B-A$ — set difference is NOT commutative, exactly like number subtraction.
If $B \subset A$: shortcut — $A \cup B = A$ and $A \cap B = B$. Skip computing, just write the answer.
$A-B$, $A\cap B$, $B-A$ are always mutually disjoint pieces that together make up $A \cup B$.

Module 6

Complement & De Morgan's Laws

$A'$ $(A')'=A$ $\varnothing'=U$ $U'=\varnothing$

$(A \cup B)' = A' \cap B'$ · $(A \cap B)' = A' \cup B'$ — the operator FLIPS

"Complement flips the operator" — the single most-flipped formula in board exams. Union becomes intersection, every time.
Forgot the law mid-exam? Just compute directly: $A' = U - A$.