This text is intended as an introduction to mathematical proofs for
students. It is distilled from the lecture notes for a course
focused on set theory subject matter as a means of teaching proofs.
Chapter 1 contains an introduction and provides a brief summary of
some background material students may be unfamiliar with. Chapters
2 and 3 introduce the basics of logic for students not yet familiar
with these topics. Included is material on Boolean logic,
propositions and predicates, logical operations, truth tables,
tautologies and contradictions, rules of inference and logical
arguments. Chapter 4 introduces mathematical proofs, including
proof conventions, direct proofs, proof-by-contradiction, and
proof-by-contraposition. Chapter 5 introduces the basics of naive
set theory, including Venn diagrams and operations on sets. Chapter
6 introduces mathematical induction and recurrence relations.
Chapter 7 introduces set-theoretic functions and covers injective,
surjective, and bijective functions, as well as permutations.
Chapter 8 covers the fundamental properties of the integers
including primes, unique factorization, and Euclid's algorithm.
Chapter 9 is an introduction to combinatorics; topics included are
combinatorial proofs, binomial and multinomial coefficients, the
Inclusion-Exclusion principle, and counting the number of
surjective functions between finite sets. Chapter 10 introduces
relations and covers equivalence relations and partial orders.
Chapter 11 covers number bases, number systems, and operations.
Chapter 12 covers cardinality, including basic results on countable
and uncountable infinities, and introduces cardinal numbers.
Chapter 13 expands on partial orders and introduces ordinal
numbers. Chapter 14 examines the paradoxes of naive set theory and
introduces and discusses axiomatic set theory. This chapter also
includes Cantor's Paradox, Russel's Paradox, a discussion of
axiomatic theories, an exposition on Zermelo-Fraenkel Set Theory
with the Axiom of Choice, and a brief explanation of Goedel's
Incompleteness Theorems.
General
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