Prep for GRE® Subject Math Exam-Module7:Additional Topics

Explore concise topics from graph theory and set theory to topology and real and complex analysis, with examples on center of gravity, least squares regression, and bisection and Newton-Raphson methods.

Explore set theory basics—unions, intersections, complements, Morgan laws, cartesian products, subsets, and mappings—plus cardinality, Cantor’s theorem, and the continuum; then study logic, truth tables, and tautologies.

Explore core ideas in combinatorics and probability, including counting methods, permutations and combinations, bars and stars, and key probability concepts like independent events, random variables, and distributions.

Explore topology concepts from open sets and bases to continuity and compactness. Learn about Hausdorff, subspace and product topologies, and the Bolzano-Weierstrass criterion for compactness.

Explore complex analysis basics: Cartesian and polar representations, conjugates, roots of unity, complex logs and powers, holomorphic, analytic and meromorphic functions, poles, and Laurent series.

Explore complex analysis: evaluate line integrals on smooth curves using complex primitives and Laurent series, including the a_{-1} coefficient, and apply residues and the residue theorem for poles inside contours.

Explore real analysis and measure theory on the real line, covering measurable sets, outer measures, the Lebesgue integral, and simple and step functions.

Tackle the final quiz with a few questions, each allotted 2.5 minutes; pace yourself, and note that answers are in the next lecture.

Explore the measurability of functions, including non-measurable cases and absolute value. Assess closures, open sets, measure zero via countable unions, distribution functions, and graph edge counts.

Complete this bonus lecture as part of the eight-module GRE subject math prep course, emphasizing practice exams, study plans, handouts, and staying updated via the course website.