maths.free › Set Theory & Logic

Set Theory & Logic

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

Уроки

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.

Оқулықтар

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

Оның кейбір сабақтары ашық мәтіннен алынған: OpenStax Contemporary Mathematics, Wikipedia. Әрбір сабақ парағының өзінің көзі мен лицензиясы көрсетілген.

Мұнда қолданылатын символдар

Тұтас анықтамасын, суретін және әрбір әріптің мағынасын көру үшін кез келген символды түртіңіз.

Адамдар қоятын сұрақтар

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.

Өзіңіздің проблемаңыз бар ма?
Оны шешу парағының бетіне жазып не суреттеп қойыңыз: әрбір қадамы сурет, сұрағанша жауап жасырылады. & Шегіну →

Басқа тармақтар

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers