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Set Theory & Logic

Sets, relations, functions, cardinality, proof and the foundations of mathematics.

Lessons

Introductory Sets and operations Membership, subsets, union, intersection, complement, and Venn diagrams. {1,2,3,4} union {3,4,5,6} Core Relations, functions and equivalence Relations as sets of pairs; reflexive, symmetric, transitive; functions and bijections. truth table of p implies q Advanced Cardinality and infinity Countable and uncountable sets, Cantor's diagonal argument, the continuum. sum of 1/2^n for n = 0 to oo Core Logic and methods of proof Connectives, quantifiers, direct proof, contrapositive, contradiction, induction. truth table of (p implies q) iff (not p or q)

What mathematics is built on. Sets and the operations on them, relations and functions defined precisely, the sizes of infinity, and the logic that turns an argument into a proof.

More lessons

Sets

Basic Set Concepts Subsets Understanding Venn Diagrams Set Operations with Two Sets Set Operations with Three Sets Set theory Set (mathematics)Subset Union (set theory)Intersection (set theory)Complement (set theory)Power set Cartesian product Empty set Venn diagram

How a proof is made

Mathematical proof Direct proof Proof by contradiction Proof by contrapositive Mathematical induction Constructive proof Proof by exhaustion Counterexample Conjecture Theorem Axiom Lemma (mathematics)Corollary Well-ordering principle Reductio ad absurdum Euclid's theorem Cantor's diagonal argument

Relations and functions

Binary relation Equivalence relation Partial order Function (mathematics)Bijection Injective function Surjective function Inverse function Composite function

Infinity

Cardinal number Ordinal number Countable set Uncountable set Continuum hypothesis Axiom of choice Zermelo–Fraenkel set theory Russell's paradox Well-order Transfinite induction

Logic

Mathematical logic Propositional calculus First-order logic Truth table Logical connective Quantifier (logic)Predicate (mathematical logic)Gödel's incompleteness theorems Model theory Proof theory Computability theory Turing machine Halting problem Peano axioms Boolean algebra

Some of these lessons are adapted from open texts: OpenStax Contemporary Mathematics, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

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