Proofs and Fundamentals

Part 1 Proofs: informal logic, introduction, statements, relations between statements, valid arguments quantifiers; strategies for proofs; mathematical proofs - what they are and why we need them; direct proofs, proofs by contrapositive and contradiction; cases, and if and only if, quantifiers in theorems, writing mathematics. Part 2 Fundamentals: sets, introduction, sets - basic definitions; set operations, indexes families of sets; functions, image and inverse image, composition and inverse functions, injectivity, surjectivity and bijectivity, set of functions; relations, congruence, equivalence relations; infinite and finite sets, cardinality of sets, cardinality of the number systems, mathematical induction, recursion. Part 3 Extras: selected topics, binary operations, groups, homomorphisms and isomorphism, partially ordered sets, lattices, counting products and sums, counting - permutations and combinations; number systems, back to the beginning, the natural numbers, further properties of the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers. Appendix: properties of numbers. Hints of selected exercises.