This is an introduction to proofs book for the course offering a transition to more advanced mathematics. It contains logic, sets, functions, relations, the construction of rational, real and complex numbers and their properties. It also has a chapter on cardinality and a chapter on counting techniques. The book explains various proof techniques and has many examples which help with the transition to more advanced classes like Real Analysis, Groups, rings and fields or topology.
This is an introduction to proofs book for the course offering a transition to more advanced mathematics. It contains logic, sets, functions, relations, the construction of rational, real and complex numbers and their properties. It also has a chapter on cardinality and a chapter on counting techniques. The book explains various proof techniques and has many examples which help with the transition to more advanced classes like Real Analysis, Groups, rings and fields or topology.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Valentin Deaconu teaches at University of Nevada, Reno.
Inhaltsangabe
Elements of logic True and false statements Logical connectives and truth tables Logical equivalence Quantifiers Proofs: Structures and strategies Axioms, theorems and proofs Direct proof Contrapositive proof Proof by equivalent statements Proof by cases Existence proofs Proof by counterexample Proof by mathematical induction Elementary Theory of Sets. Functions Axioms for set theory Inclusion of sets Union and intersection of sets Complement, difference and symmetric difference of sets Ordered pairs and the Cartersian product Functions Definition and examples of functions Direct image, inverse image Restriction and extension of a function One-to-one and onto functions Composition and inverse functions *Family of sets and the axiom of choice Relations General relations and operations with relations Equivalence relations and equivalence classes Order relations *More on ordered sets and Zorn's lemma Axiomatic theory of positive integers Peano axioms and addition The natural order relation and subtraction Multiplication and divisibility Natural numbers Other forms of induction Elementary number theory Aboslute value and divisibility of integers Greatest common divisor and least common multiple Integers in base 10 and divisibility tests Cardinality. Finite sets, infinite sets Equipotent sets Finite and infinite sets Countable and uncountable sets Counting techniques and combinatorics Counting principles Pigeonhole principle and parity Permutations and combinations Recursive sequences and recurrence relations The construction of integers and rationals Definition of integers and operations Order relation on integers Definition of rationals, operations and order Decimal representation of rational numbers The construction of real and complex numbers The Dedekind cuts approach The Cauchy sequences approach Decimal representation of real numbers Algebraic and transcendental numbers Comples numbers The trigonometric form of a complex number
Elements of logic True and false statements Logical connectives and truth tables Logical equivalence Quantifiers Proofs: Structures and strategies Axioms, theorems and proofs Direct proof Contrapositive proof Proof by equivalent statements Proof by cases Existence proofs Proof by counterexample Proof by mathematical induction Elementary Theory of Sets. Functions Axioms for set theory Inclusion of sets Union and intersection of sets Complement, difference and symmetric difference of sets Ordered pairs and the Cartersian product Functions Definition and examples of functions Direct image, inverse image Restriction and extension of a function One-to-one and onto functions Composition and inverse functions *Family of sets and the axiom of choice Relations General relations and operations with relations Equivalence relations and equivalence classes Order relations *More on ordered sets and Zorn's lemma Axiomatic theory of positive integers Peano axioms and addition The natural order relation and subtraction Multiplication and divisibility Natural numbers Other forms of induction Elementary number theory Aboslute value and divisibility of integers Greatest common divisor and least common multiple Integers in base 10 and divisibility tests Cardinality. Finite sets, infinite sets Equipotent sets Finite and infinite sets Countable and uncountable sets Counting techniques and combinatorics Counting principles Pigeonhole principle and parity Permutations and combinations Recursive sequences and recurrence relations The construction of integers and rationals Definition of integers and operations Order relation on integers Definition of rationals, operations and order Decimal representation of rational numbers The construction of real and complex numbers The Dedekind cuts approach The Cauchy sequences approach Decimal representation of real numbers Algebraic and transcendental numbers Comples numbers The trigonometric form of a complex number
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