Download e-book for iPad: Mathematics : A Minimal Introduction by Alexandru Buium

By Alexandru Buium

ISBN-10: 1482216019

ISBN-13: 9781482216011

Entrance hide; Contents; Preface; advent; half 1. Pre-mathematical good judgment; bankruptcy 1. Languages; bankruptcy 2. Metalanguage; bankruptcy three. Syntax; bankruptcy four. Semantics; bankruptcy five. Tautologies; bankruptcy 6. Witnesses; bankruptcy 7. Theories; bankruptcy eight. Proofs; bankruptcy nine. Argot; bankruptcy 10. recommendations; bankruptcy eleven. Examples; half 2. arithmetic; bankruptcy 12. ZFC; bankruptcy thirteen. units; bankruptcy 14. Maps; bankruptcy 15. Relations;

Chapter 24. ImaginariesChapter 25. Residues; bankruptcy 26. p-adics; bankruptcy 27. teams; bankruptcy 28. Orders; bankruptcy 29. Vectors; bankruptcy 30. Matrices; bankruptcy 31. Determinants; bankruptcy 32. Polynomials; bankruptcy 33. Congruences; bankruptcy 34. traces; bankruptcy 35. Conics; bankruptcy 36. Cubics; bankruptcy 37. Limits; bankruptcy 38. sequence; bankruptcy 39. Trigonometry; bankruptcy forty. Integrality; bankruptcy forty-one. Reciprocity; bankruptcy forty two. Calculus; bankruptcy forty three. Metamodels; bankruptcy forty four. different types; bankruptcy forty five. Functors; bankruptcy forty six. goals; half three. Mathematical common sense; bankruptcy forty seven. types; bankruptcy forty eight. Incompleteness.

Pre-Mathematical good judgment Languages Metalanguage Syntax Semantics Tautologies Witnesses Theories Proofs Argot techniques Examples arithmetic ZFC units Maps family members Operations Integers Induction Rationals Combinatorics Sequences Reals Topology Imaginaries Residues p-adics teams Orders Vectors Matrices Determinants Polynomials Congruences strains Conics Cubics Limits sequence Trigonometry Integrality Reciprocity Calculus Metamodels different types Functors pursuits Mathematical good judgment types Incompleteness Bibliography Index. Read more...

summary: entrance conceal; Contents; Preface; advent; half 1. Pre-mathematical good judgment; bankruptcy 1. Languages; bankruptcy 2. Metalanguage; bankruptcy three. Syntax; bankruptcy four. Semantics; bankruptcy five. Tautologies; bankruptcy 6. Witnesses; bankruptcy 7. Theories; bankruptcy eight. Proofs; bankruptcy nine. Argot; bankruptcy 10. innovations; bankruptcy eleven. Examples; half 2. arithmetic; bankruptcy 12. ZFC; bankruptcy thirteen. units; bankruptcy 14. Maps; bankruptcy 15. family members; bankruptcy sixteen. Operations; bankruptcy 17. Integers; bankruptcy 18. Induction; bankruptcy 19. Rationals; bankruptcy 20. Combinatorics; bankruptcy 21. Sequences; bankruptcy 22. Reals; bankruptcy 23. Topology.

Chapter 24. ImaginariesChapter 25. Residues; bankruptcy 26. p-adics; bankruptcy 27. teams; bankruptcy 28. Orders; bankruptcy 29. Vectors; bankruptcy 30. Matrices; bankruptcy 31. Determinants; bankruptcy 32. Polynomials; bankruptcy 33. Congruences; bankruptcy 34. strains; bankruptcy 35. Conics; bankruptcy 36. Cubics; bankruptcy 37. Limits; bankruptcy 38. sequence; bankruptcy 39. Trigonometry; bankruptcy forty. Integrality; bankruptcy forty-one. Reciprocity; bankruptcy forty two. Calculus; bankruptcy forty three. Metamodels; bankruptcy forty four. different types; bankruptcy forty five. Functors; bankruptcy forty six. goals; half three. Mathematical common sense; bankruptcy forty seven. versions; bankruptcy forty eight. Incompleteness.

Pre-Mathematical good judgment Languages Metalanguage Syntax Semantics Tautologies Witnesses Theories Proofs Argot thoughts Examples arithmetic ZFC units Maps kinfolk Operations Integers Induction Rationals Combinatorics Sequences Reals Topology Imaginaries Residues p-adics teams Orders Vectors Matrices Determinants Polynomials Congruences strains Conics Cubics Limits sequence Trigonometry Integrality Reciprocity Calculus Metamodels different types Functors pursuits Mathematical good judgment types Incompleteness Bibliography Index

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Extra info for Mathematics : A Minimal Introduction

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A definition in L is a sentence of one of the following types: 1) “c = t” where c is a constant and t is a term without variables. 2) “∀x( (x) ↔ E(x))” where is a unary relational predicate and E is a formula with one free variable. More generally for several variables, “∀x∀y( (x, y) ↔ E(x, y))” is a definition, etc. 3) “∀x∀y((y = f (x)) ↔ F (x, y))” where f is a unary functional predicate and F is a formula with 2 free variables; more generally one allows several variables. If any type of symbols is missing from the language we disallow, of course, the corresponding definitions.

And lists of words in each grammatical category. Then one defines a noun phrase (NP) as a noun possibly preceded by adjectives and a determinator and possibly followed by a prepositional phrase (PP); one defines propositional phrases, verb phrases, etc. in a similar way. This scheme can be formalized. What results is a very general theory applicable to virtually all natural languages, not only to English. This kind of grammatical syntax differs from the logical syntax explained before this remark and is not appropriate for introducing mathematics; therefore we will not pursue this grammatical syntax further.

Hint: ((S → S ) ∧ (S → S )) → (S → S ) is a tautology. 22. Explain why if the sentences S , S appear in a proof then adding S ∧ S to the proof yields a proof. 23. Explain why if S appears in a proof and S → S is a tautology then adding S to the proof yields a proof. 24. Explain why if S and S → S appear in the proof then adding S to the proof yields a proof. 25. Explain why if P → R and Q → R appear in a proof then adding (P ∨ Q) → R to the proof yields a proof. 26. , either P is in T or ¬P is in T .

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Mathematics : A Minimal Introduction by Alexandru Buium


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