Set Theory Introduction
The Language of Mathematics
Introduction (high school to first-year university)
About This Level
At the introductory level we come to understand the notion of a “set” as the basic language of mathematics. A set is more than a mere collection of things: it is the tool with which mathematical objects are described in a uniform way. Through set operations, mappings and relations, the goal is to get ready to speak mathematics.
Prerequisites
- High-school mathematics
- The ability to read a logical argument
- A willingness to generalise from concrete examples
Contents
1. What Is a Set?
The starting point of naive set theory.
- The intuitive definition of a set
- Elements and membership $\in$
- Notation for sets (roster and set-builder)
2. Subsets and Equality
Comparing sets.
- Subsets $\subseteq$
- Equality of sets
- The empty set $\emptyset$
3. Set Operations
Building new sets.
- Union $A \cup B$
- Intersection $A \cap B$
- Difference and complement
4. Mappings (Functions)
Correspondences between sets.
- The definition of a mapping
- Domain, image and codomain
- Surjections, injections and bijections
5. Composition and Inverse Mappings
Operating on mappings.
- The composite $g \circ f$
- When an inverse mapping exists
- The identity mapping
6. Relations
How elements relate to one another.
- Binary relations
- Equivalence relations
- Quotient sets
Related Terms (Glossary)
Venn Diagram
Visualising set operations and the inclusion–exclusion principle.
De Morgan's Laws
The duality that swaps union and intersection under complementation.
Power Set
The set of all subsets of a set; $2^n$ of them when the set is finite, leading in general to Cantor's theorem.
Symmetric Difference
Elements belonging to exactly one side, and the correspondence with XOR.
Equivalence Relation
Reflexivity, symmetry, transitivity, and quotient sets.
Next Step
Once the basics of sets are in place, move on to Basic Level, Chapter 1: start from power sets and Cartesian products, then study the theory of cardinality, countable and uncountable sets, and the diagonal argument.
Key Concepts
The intuitive notion of a set
In naive set theory a set is understood intuitively as “several objects taken together”. For an element $x$ of a set $A$ we write $x \in A$. Not every property, however, may be used to collect objects into a set without restriction.
Definition of a mapping
A mapping $f: A \to B$ from a set $A$ to a set $B$ assigns to each element $a$ of $A$ exactly one element $f(a)$ of $B$.
Definition of an equivalence relation
A relation $\sim$ on a set $A$ is an equivalence relation when it satisfies: (1) reflexivity, $a \sim a$; (2) symmetry, $a \sim b \Rightarrow b \sim a$; (3) transitivity, $a \sim b, b \sim c \Rightarrow a \sim c$.
Bijections and cardinality
When a bijection exists between two sets $A$ and $B$, they are said to have the same cardinality. This is the rigorous definition of “the same size”.
Paradoxes of Naive Set Theory
Russell's paradox
Assuming that “the set of all sets that do not contain themselves as an element” exists leads to a contradiction. Put $R = \{x : x \notin x\}$ and ask whether $R \in R$: the answer breaks down either way.
The motivation for axiomatic set theory
To avoid the paradoxes, which sets are taken to exist must be restricted by axioms. This is the road that leads to the ZFC axiom system.
What You Can Understand at This Level
The basics of mathematical writing
You become able to read and write notation such as “$\forall x \in A$” and “$\exists y \in B$” accurately.
A rigorous grasp of functions
The “function” met at school is redefined as a mapping, and the importance of domain and range becomes clear.
Thinking in equivalence classes
New mathematical objects are built from a relation of “regarding as the same”. This prepares the construction of the integers and the rationals.
The gateway to infinity
The set of all natural numbers is infinite. You gain an intuitive grasp of how it differs essentially from a finite set.
Study Tips
- Get used to the symbols: use $\in$, $\subseteq$, $\cup$, $\cap$ and the rest until they feel natural
- Make your own examples: check every abstract definition against a concrete set
- Keep definitions exact: do not leave “mapping” or “surjective” vague
- Learn from the paradoxes: understand what goes wrong with the naive notion of a set, and why axiomatic set theory became necessary
Indexes
Definition Index (General Concepts)
A list of the terms, symbols and functions defined at this level.
53 entries Symbol Index
41 entries Function Index
5 entries Name Index
3 people Theorems and Proofs
17 entries
Worked Examples (This Level Only)
A list of concrete numerical and computational examples.
Reading Pieces
Collect Everything and It Breaks [Reading]
“A collection of things” was not enough. From the riddle of the village barber to Russell's paradox — a relaxed account of the moment the naive idea that any condition defines a set collapsed in a single line.
What Three Circles Tell Us [Reading]
Narrowing a search, double-counting in a questionnaire, swapping “not” around. Taking overlapping circles — Venn diagrams — as a guide, a light-hearted look at where union, intersection and difference hide in daily life.
Everything Starts from a Collection [Reading]
Numbers, points and functions can all be written with sets if you trace them far enough back. Why could a single phrase, “a collection of things”, become the common language underpinning the whole of mathematics? From Cantor to the construction of the natural numbers, told at an easy pace.