mathematize

a program for inducing mathematical excellence

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These are lecture notes for a short course in proof-based mathematics, designed for a bright high schooler with no mathematical exposure past precalculus. We aim to introduce the student to the language of modern mathematics, initially beginning with a primer on traditional set-theoretical language and concepts, before surveying basic topics in algebra, analysis, and ending with a category theoretical viewpoint on all topics covered heretofore.

Topics we aim to cover include elements of propositional logic, mathematical proof, fundamental mathematical structures (sets, symmetry groups, dihedral groups, cyclic groups), fundamental concepts in analysis on the real line (neighborhoods, sequences, limits), basic concepts in counting and probability (combinatorial methods, conditioning and Baye's Theorem), and categorical concepts such as categories, morphisms, isomorphisms, functors, and monads.