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The Penguin's Field Guide to Sets, Functions & Proofs

The Penguin's Field Guide to Sets, Functions & Proofs

The Penguin's Field Guide

Sets · Functions · Sequences · Proofs

🐧 survival kit

Jump To

01 Set Basics
02 Set Operations
03 Power Sets
04 Cartesian Products
05 Functions
06 Floor & Ceiling
07 Sequences & Sums
08 Proof Techniques
09 Cheat Sheet

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Set Basics

📐 What is a Set?

An unordered collection of distinct objects. Order doesn't matter, duplicates collapse.

\(\{1, 2\} = \{2, 1\}\)  ·  \(\{1, 1, 2\} = \{1, 2\}\)

🔢 Counting Elements — Watch the Nesting

Only count top-level elements. Nested sets are single objects.

\(\{0,\ \{1, \{2,3\}\}\}\) has 2 elements: the number \(0\) and the set \(\{1,\{2,3\}\}\)

Don't unpack nested sets. A box containing boxes still counts as one item on the shelf.

∅ vs {∅} — The #1 Trap

This distinction is everywhere on this test.

Symbol
What It Is
Elements

\(\emptyset\)
The empty set
0 elements

\(\{\emptyset\}\)
A set containing the empty set
1 element

Think of \(\emptyset\) as an empty bag. \(\{\emptyset\}\) is a bag with an empty bag inside it — it's not empty.

∈ vs ⊆ — Element vs Subset

Notation
Meaning
Example

\(a \in S\)
\(a\) is an element inside \(S\)
\(1 \in \{1,2\}\) ✓

\(A \subseteq S\)
Every element of \(A\) is in \(S\)
\(\{1\} \subseteq \{1,2\}\) ✓

Memorize
\(\emptyset \subseteq S\) for every set \(S\) (vacuously true — nothing to check).

Memorize
\(\emptyset \in S\) is true only if \(S\) explicitly contains \(\emptyset\) as an element.

Trap
\(\emptyset \in \emptyset\) is false — the empty set has no elements at all.

Subset vs Proper Subset

\(A \subseteq B\): every element of \(A\) is in \(B\)
\(A \subset B\): \(A \subseteq B\) and \(A \neq B\)

Memorize
Every set is a subset of itself: \(A \subseteq A\). No set is a proper subset of itself.

Number Sets Hierarchy

\(\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)

\(\mathbb{Q}\) (rationals) is a proper subset of \(\mathbb{R}\) (reals) because irrationals exist. \(\mathbb{R}\) is NOT a subset of \(\mathbb{Q}\).

Interval Notation

Notation
Meaning
Endpoints

\([a, b]\)
\(a \leq x \leq b\)
Both included

\((a, b)\)
\(a < x < b\)
Both excluded

\([a, b)\)
\(a \leq x < b\)
Left in, right out

\((a, b]\)
\(a < x \leq b\)
Left out, right in

\([1,2]\) is a set. "\(1 \leq x \leq 2\)" is a statement.…