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Expanded Course Description
The discipline of mathematics includes two related strands: calculation (the main avenue of application of mathematics) and proof (the means by which mathematical truth is verified, and by which calculational algorithms are shown to be correct). Most of our math courses through the 200-level are devoted to calculation (and to applications of calculation). By contrast, this course is devoted to teaching techniques of mathematical proof and formalism in the context they naturally arise for working mathematicians: while trying to understand interesting problems.
By teaching students about how to do and think about proof, this course introduces students to the foundation of mathematics as a discipline (and thus is a science group satisfying course). For math majors (or minors) this course can also serve as a bridge between the calculation and application based courses that most students take at the 100- and 200-level in mathematics and the more theoretical content of many 300- and 400-level courses.
This course will address fundamental questions such as
-"What is a proof and how are basic proofs constructed?"
-"What is mathematical induction, and what different kinds of proofs are there?"
-"How does one understand and use logical constructions such as converse and contrapositive?"
-"What does it mean to prove some fact which is already familiar and believable?"
-"How does one start with an interesting problem, work through examples to form a strategy for a proof capturing the essence of the problem?"
The specific mathematical topics used to address these questions include:
-the meaning (to a mathematician) of equations and sets,
-the relationship between set theoretic and logical properties (for example between inclusion of sets and logical implication),
-using induction in a wide array of settings (including establishing equalities and inequalities and analyzing "games"),
-counting basic structures such as subsets and permutations,
-basic properties of numbers (such as unique factorization of integers into primes, or ability to divide with remainder),
-Fermat's Little Theorem and its usage in cryptography,
-the analysis of graphs and associated structures (orientations and colorings) and how they can be used to model problems.