EITHER:
For Math I : Discrete Mathematics:
Logic: propositional logic; predicates and
quantifiers; rules of inference.
The integers: basic properties of the integers ,
divisibility ; prime numbers.
Mathematical reasoning: methods of proof, direct
proof and indirect proof. Mathematical induction.
Sets and set operations . Power set and cardinality
of sets. Proof using set identities.
Functions: definition and properties of functions.
Composition of functions. Injections, surjections,
and inverses. Exponentials , logarithms, factorials ,
polynomials.
Summations: arithmetic and geometric
progressions. Finding closed forms.
Relations: properties of binary relations.
Composition and closures. Equivalence relations
and partial orders. Matrices.
Modular arithmetic: basic arithmetic of the integers
mod p.
Graphs: definition of graphs , paths, trees.
For Math II: Counting and Discrete Probability:
Growth of functions: asymptotic notation and
asymptotic order of functions .
Recursive definitions, recursively defined
functions, Fibonacci numbers.
Recurrences: methods of solving recurrences .
Counting: basic of counting; permutations,
combinations, binomial theorem.
Discrete probability: probability distribution ;
conditional probability and independence; random
variables and expectation; variance. Statistics:
mean, variance, and standard deviation, normal
distribution, geometric and binomial distributions.
OR
Students, who have a B.A. or a M.A. degree in
mathematics, are permitted to place out of
immersion mathematics. Other students who have
taken higher-level mathematics courses may
petition for placement.
There are some topics in calculus and algebra that
are relevant to discrete mathematics; they are listed
here.
Calculus topics: basic properties of numbers, proof
by induction; functions, limits, continuous
functions, inverse functions, least upper bounds;
logarithm and exp onential functions , polynomial
functions; infinite sequences, infinite series.
Note: the calculus course should have exposed the
student to proofs and should not be limited to
solving simple problems (e.g., finding derivatives
and integrals by applying a rule).
Algebra topics: integers, divisors, prime numbers,
congruences; functions, equivalence relations,
permutations; polynomials, roots of polynomials ;
matrices and linear algebra .
Note: the standard topics in abstract algebra
courses are groups , rings, fields, vector spaces,
linear algebra, and polynomials in several
variables. Number theory and matrices are the only
topics that are directly relevant to discrete
mathematics.
