Schröder–Bernstein theorem

Definition

In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.

Related concepts

Abductive reasoning Abraham Fraenkel Abstract logic Ackermann set theory Aleph number Algebraic logic Almost Alphabet (formal languages)Alternative set theory Alwin Reinhold Korselt Amorphous set Analytic–synthetic distinction Antecedent (logic)Antinomy Argument Argumentation theory Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of adjunction Axiom of choice Axiom of constructibility Axiom of countable choice Axiom of dependent choice Axiom of determinacy Axiom of extensionality Axiom of global choice Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of power set Axiom of projective determinacy Axiom of regularity Axiom of separation Axiom of union Axiom schema Axiom schema of replacement Axiom schema of specification Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Burali-Forti paradox Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinal number Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Citizendium Class (set theory)Classical logic Codomain Compactness theorem Complement (set theory)Complete theory Computability theory Computable function Computable set Computably enumerable set Concrete category Consequent Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Contradiction Corollary Countable set Critical thinking De Morgan's laws Decidability (logic)Decision problem Dedekind-infinite set Deductive closure Deductive reasoning Deductive system Definition Description Diagram (mathematical logic)Dichotomy Disjoint set Disjoint union Domain of a function Doubly infinite Egbert Brieskorn Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equipotent Equivalence relation Eric W. Weisstein Ernst Schröder (mathematician)Ernst Zermelo Euclid's Elements Euclidean geometry Excluded middle Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Family of sets Felix Bernstein (mathematician)Felix Hausdorff Filter on a set Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal semantics (natural language)Formal system Formation rule Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Friedrich Moritz Hartogs Function (mathematics)Functional predicate Fuzzy set General set theory Georg Cantor Grothendieck universe Ground expression Ground formula Gunter M. Ziegler Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hereditarily finite set Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Index of logic articles Inductive reasoning Inference Infinite-valued logic Infinite set Informal logic Information theory Inhabited set Injective function International Congress of Mathematicians Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionism Isomorphism Jahresbericht der Deutschen Mathematiker-Vereinigung John von Neumann Julius König Knaster–Tarski theorem Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lemma (mathematics)Lindström's theorem Linguistic entailment List of Boolean algebra topics List of Hilbert systems List of axioms List of fallacies List of first-order theories List of formal systems List of logic symbols List of logicians List of mathematical logic topics List of mathematical theories List of paradoxes List of rules of inference List of set identities and relations List of set theory topics List of statements independent of ZFC Logic Logic in computer science Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical form Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Martin's axiom Martin Aigner Material conditional MathWorld Mathematical Proceedings of the Royal Irish Academy Mathematical logic Mathematical object Mathematische Annalen Metalanguage Metalogic Metamathematics Minimal axioms for Boolean algebra Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Myhill isomorphism theorem NLab NP (complexity)Naive set theory Name Natural deduction Necessity and sufficiency Negation Netto's theorem New Foundations Non-Euclidean geometry Non-classical logic Non-logical symbol Non-standard model Non-standard model of arithmetic Open formula Operation (mathematics)Ordered pair Ordinal analysis Ordinal number Otto Blumenthal Outline of logic P (complexity)P versus NP problem Paradox Paradoxes of set theory Partial function Partition of a set Patrick Suppes Paul Bernays Paul Cohen Peano axioms Philosophical logic Philosophy of logic Philosophy of mathematics Power set Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Probability Proof of impossibility Proof theory Proofs from THE BOOK Proposition Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Reason Recursion Recursive set Reference Relation (mathematics)Reverse mathematics Richard Dedekind Robert Fricke Robinson arithmetic Rule of inference Russell's paradox Satisfiability Saturated model Schröder–Bernstein property Schröder–Bernstein theorem for measurable spaces Schröder–Bernstein theorems for operator algebras Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set-builder notation Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Springer books Square of opposition Statement (logic)Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Subset Substitution (logic)Substructure (mathematics)Supertask Surjective function Suslin's problem Syllogism Symbol (formal)Symmetric difference Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic Theorem Theories of truth Theory (mathematical logic)Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Transfer principle Transfinite induction Transitive set True arithmetic Truth Truth predicate Truth table Truth value Tuple Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultrafilter on a set Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Walther von Dyck Well-formed formula Well-ordering theorem Willard Van Orman Quine William Stanley Jevons Zeitschrift für Philosophie und philosophische Kritik Zermelo set theory Zermelo–Fraenkel set theory Émile Borel

152 concepts already in your glossary