IMACS Afterschool Programs

the substitution principle and the tautology principle; Modus Ponens, conjunctive inference and conjunctive simplification, contrapositive inference and Modus Tollens, syllogistic inference and inference by cases; the general substitution principle; the Deduction Theorem and the principle of indirect inference; the object language/metalanguage distinction; an introduction to quantification; informal treatments of set membership and equality; the empty set; Venn diagrams; subsets; power sets; Pascals formula for the number of k-element subsets of an m-element set; the intersection, union, and difference of sets; Cartesian products of sets; componentwise operations; terms (including class symbols) and well-formed formulas; free and bound occurrences of variables, the dilemma of the bound variable; the principle of generalization to a universal, inference from a universal; the axiom of extensionality and the comprehension principle; properties of equality; inference from an existential, the principle of generalization to an existential; starting the transition from demonstrations to paragraph proofs; the Deduction Theorem and the general substitution principle for the predicate calculus; an introduction to modified Zermelo set theory; relations and functions; binary operations; manifold union and intersection; the axiom of separation; Russell's Paradox; the axiom of pure sets; a general discussion of first-order logics (with equality). Classes meet 2 hours per week.

Note: This is not the same course as the Logic for Mathematics online course provided by eIMACS. The latter is a simplified version of the first section of the Logic for Mathematics course described above.

Cayley's Theorem; counting techniques, permutations and combinations, factorials, binomial coefficients, Pascal's Triangle, the product rule; the Binomial Theorem; unordered sums and products over finite sets; matrices and matrix algebra; using matrices to solve systems of linear equations; the ring of functions from one set to another; the ring of polynomial functions over a field; the Remainder Theorem and the Factor Theorem; synthetic division; the Rational Root Theorem. Classes meet 4 hours per week, with significant amounts of homework.

Numeration systems; base m positional and periodic names for rational numbers (focusing primarily on the case when m = 10); decimal approximation sequences for real numbers. Convergence of real sequences treated from a general metric space point of view; the equivalence for real sequences between convergence and satisfying the Cauchy condition, the Bolzano-Weierstrass Theorem; a short introduction to the convergence of series. Real powers; exponential and logarithmic functions and their algebraic properties. Completeness, and the uniqueness of the real number system. Applications of the representation in the complex plane of sets of complex numbers; polynomial functions over the complex field. An introduction to equipollence and dominance in preparation for the study in "Elements of Mathematics" Book 8 of cardinal numbers; Cantor's Theorem; the Schröder-Bernstein Theorem; the Cantor discontinuum. Classes meet 4 hours per week, with significant amounts of homework.