maths.free › Set Theory & Logic

Set Theory & Logic

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

레슨

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.

더 많은 레슨

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

이러한 교훈 중 일부는 개방된 텍스트에서 적용됩니다. OpenStax Contemporary Mathematics, Wikipedia. 각 레슨 페이지는 소스와 라이선스를 명시합니다.

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

사람들이 묻는 질문

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.

자신의 문제가 있습니까?
솔버 페이지에 답을 입력하거나 그려보세요. 모든 단계, 그림, 답은 질문할 때까지 숨겨져 있습니다. 해결 →

다른 지점

＋ 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