## Discrete Mathematics: Math 245, Spring 2012 Schedule

I will try to keep to this schedule but will update it as needed.

Solutions to last HW.
Permutations and combinations.
Pascal's triangle.
Day Topics Preparation
Th. 1/19 Introduction. Statements, negation, conjunction, disjunction.
Logical equivalences
Sec. 2.1
Tu. 1/24 Logical equivalences, tautologies, contradictions.
Conditionals and biconditionals.
Secs. 2.1-2
Th. 1/26 Conditionals and English usage.
Logical arguments.
Secs. 2.2-3
Tu. 1/31 Logic circuits and logic puzzles.
Sets
Secs. 2.4, 1.1-2, 6.1-2 (pp.336-344)
Th. 2/2 Sets, complement, intersection, union.
Main properties of sets.
Predicates and truth sets
Secs. 6.1-2, 3.1-2
Tu. 2/7 Predicates and English.
Quantified predicates. Negation.
Multiply quantified predicates.
Secs. 3.1-3
Th. 2/9 Arguments with quantified predicates: Puzzles
Proofs of set properties: element-wise.
Sec. 3.4
Tu. 2/14 Algebraic proofs of set theorems. Disproof by counterexample.
Sets: Cartesian products and the multiplication rule.
Secs. 6.1,2
Th. 2/16 Sets and counting:
Partitions and inclusion/exclusion.
Power sets
Secs. 6.1-3
Sec. 6.1 (p. 344-348)
Sec. 9.2 (525-529), Sec. 9.3
Tu. 2/21 Mathematics: Axiom, Definition, Theorem, Proof
Axioms for the integers.
Fundamental properties of the integers. Order.
Sec. 3.1
Th. 2/23 Integers: Divisibility, linear combinations.
The quotient-remainder theorem.
Secs. 3.1, 3.3, 3.4
Tu. 2/28 TEST: Logic, sets, counting Ch. 2, 3, 6
Th. 3/1 Applications of the quotient-remainder theorem.
The Euclidean algorithm.
Binary expansion of a number.
Secs. 4.1, 4.3,
Sec. 4.8
Tu. 3/6 Arithmetic in base b .
Primes and the Unique Factorization Theorem.
3rd E.: Sec. 1.5 (pp. 57-60, 70-73) 4.1 (pp. 211-213) 3.3 (pp. 153-4).
4th Ed.:Sec. 2.5 (pp. 78-81, 91-93), 5.1 (pp. 240-242), 4.3 (pp.176-7)
Th. 3/8 Infinitude of primes.
Rational Numbers and real numbers.
The floor and ceiling functions.
Sec. 4.5-7
Tu. 3/13 Sequences, summation and product notation.
The well ordering principle.
Secs. 5.1-2, 5.4
Th. 3/15 The principle of induction.
Proof by induction. Recursively defined functions.
Sec. 5.2,6
Tu. 3/20 Induction proofs for divisibility and order. Sec. 5.3
Th. 3/22 Strong Induction. Sec. 5.4, 6, 7
Tu. 4/3 Functions and relations. Sec. 7.1-2.
Sec. 8.1 (except last two pages)
Th. 4/5 TEST: Properties of the integers, sequences, induction. Chs. 4, 7
Tu. 4/10 Relations and their inverses.
Functions: injective (one-to-one), surjective (onto).
Sec. 8.1
Sec. 7.2
Th. 4/12 Functions: composition.
Relations on a set: reflexive, symmetric, transitive.
Sec. 7.3
Sec. 8.1 (p. 446), Sec. 8.2
Tu. 4/17 Equivalence relations and partitions.
Examples of equivalence relations.
Sec 8.3
Th. 4/19 Partially ordered sets. Sec 8.5
Tu. 4/24 Partially ordered sets.
The pigeonhole principle.
Four ways to count.
Sec. 8.5
Sec. 9.4.
Sec. 9.5
Th. 4/26 Sec. 9.6,7
Tu. 5/1 TEST: Functions, relations, equivalence relations, posets. Secs 7.1,2,4. Secs. 8.1,2,3,5.
Th. 5/3 Binomial Theorem
From counting to probability and gambling.
Poker hands.
Sec. 9.1,5
Tu. 5/8 Variety show.
Questions?
Tu. 5/10 (8:00-10:00) FINAL EXAM. Cumulative.