The document outlines topics in mathematical theory including: defining functions and relations; properties of functions such as composition, inverses, one-to-one and onto functions; properties of relations including reflexivity, symmetry and transitivity; using principles of mathematical induction to prove statements; proving statements using proof by contradiction; and closure properties of regular languages. Specific problems include proving square root of 2 is irrational, using induction to prove a summation formula, and analyzing relations.