These notes are posted as a service to you. They maybe hard to understand without attending class and hearing verbal explanations from lectures; they are not meant to replace your textbook; in particular they are not as detailed as a textbook and they have not been as carefully edited and proofread as a textbook, occasioanlly they may contain typos (if you notice a typo pleas email me so I can correct it).
The millennium prize problems You can also check the book "The Millennium problems: the seven greatest unsolved mathematical puzzles of our time" by K. Devlin. Math stacks QA93 .D485 2002
Lesson 1Chapter 1 .Definitions, proofs, the language of math. Connectives.
Lesson 2Sec 2.1 and 2.2 .Equivalent statements, truth tables, => and <=>
Lesson 3Sec 2.2, 2.3 (no sets) and 2.4. .Quantifiers. Negation.
Lesson 4Sec 1.2, 3.1,3.2,3.3 (no congruences) . Three proof methods
Lesson 5Sec 3.1,3.2,3.3, 3.5 (no congruences) . More proofs, <=>,nested quantifiers, division th
Lesson 6Sec 4.1, More proofs, Induction.
Lesson 7Sec 4.1, 4.2 More Induction.
Lesson 8Sec 4.3. Strong induction, recursive definitions. Fibonacci numbers
Lesson 9Sec 4.3, 4.4 Induction. More examples.
Lesson 10Sec 2.3, 5.1,5.2. Sets.
Lesson 11Sec 5.3,5.4. Sets proofs. Cartesian product
Lesson 12Sec 5.4, 5.1 Cartesian product. Power set.
Lesson 13Sec 6.1,6.2,6.3 Functions. Injective and surjective functions
Lesson 14Sec 6.1,6.2,6.3 Functions. More about Injective and surjective functions. Composition.
Lesson 15Sec 6.4,6.5 Inverse functions.
Lesson 16Sec 9.1,9,2 Denumerable sets.
Lesson 17Sec 9.2,9.3 More denumerable sets . [0,1] is uncountable.
Lesson 18Sec 9.3 More uncountable sets .
Lesson 19Sec 3.1 Congruences mod m
Lesson 20Sec 7.4 Computing mod m
Lesson 21Sec 7.1,7.2. Equivalence relations.
Lesson 22Sec 7.3. Equivalence classes.
Lesson 23Ch 8. Number theory premiliminaries.
Lesson 24Cancellation law. Linear Congruences
Lesson 25 Inverses
Lesson 26 Fermat s little theorem. Euler s theorem.