Topics Covered

NOTE: The topics and the schedule below are subject to change.

(1) Chapter 0: compound statements, negation, logical connectives (AND, OR, IF THEN – implies, IFF – if and only if), negation of statements with and, or, if then, iff, contrapositive, converse; existential and universal quantifiers, negation of statements containing such quantifiers, proof techniques: direct proof, by counterexample, by contrapositive, by contradiction, by cases, proof of iff, proof of “the following are equivalent”.

(2) Section 5.1: the weak and the strong  forms of the Principle of Mathematical induction.

(3) Sections 1.1 and 1.2: truth tables, propositions, logical equivalence.

(4) Section 1.3: logical arguments (resolution proofs).

(5) Sections 2.1 and 2.2: Sequences, multisets, sets and operations on them, notation for union, intersection, and Cartesian product of a finite number of sets, disjoint, pairwise disjoint, partition.

(6) Wed, Feb 28, 07: Section 2.3, 2.4, and 2.5: Binary relations: reflexive.

(7) Fri, Mar 2, 07: Binary relations continued; symmetric, antisymmetric, equivalence relations, partial orders; equivalence classes.

(8) Mon, Mar 5, 07: Continued.

(9) Wed, Mar 7, 07: Continued; comparable and incomparable elements; total orders; representation of a binary relation as a matrix and as a diagraph; notation for equivalence relations and partial orders.

(10) Thur, Mar 8, 07: Functions (Chapter 3); domain, target; range; image of an element/set; preimage of an element/set; preimage of the image of a set; image of the preimage of a set.

(11) Fri, Mar 9, 07: Functions continued.

(12) Mon, Mar 12, 07: Functions continued; one-to-one (injective) functions.

(13) Wed, Mar 14, 07: Functions continued; onto (surjective) functions; bijective (one-to-one correspondence) functions; cardinality of finite and infinite sets; the cardinality of the set of Natural Numbers.

(14) Thur, Mar 15, 07: Functions continued; composition and inverse.

(15) Fri, Mar 16, 07: Continued.

(16) Mon, Mar 26, 07: Continued.

(17) Wed, Mar 28, 07: Cardinality, countable, countably infinite, uncountable, (functions from/to higher dimensions not covered)

(18) Thur, Mar 29, 07: Continued, begin Chapter 4 (Integers).

(19) Fri, Mar 30, 07: divides, quotient, remainder, how to find them, related theorems; greatest common divisor; relatively prime.

(20) Mon, April 2, 07: Section 4.2 continued: greatest common divisor, writing gcd(a,b) as a linear combination of a and b; the least common multiple, exercises.

(21) Wed, April 4, 07: Section 4.3: prime numbers.

(22) Thur, April 5, 07: Section 4.4 and additional material: Congruence and Integers modulo n.

(23) Fri, April 6, 07: continued (finish the Chapter).

(24) Wed, April 6, 07: continued (finish the Chapter).

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(25) Friday, April 13, 07: Matrices: definitions of row vectors, column vectors, square matrices, zero matrix, identity matrix, square matrices, using matrices to solve linear systems, other topics.

(26) Monday, April 16, 07: Addition of matrices and vectors, scalar multiplication of matrices and vectors, dot/inner product of vectors, multiplication of matrices.

(27) Wednesday, April 18, 07: Powers of matrices, transpose, and special matrices.

(28) Thursday, April 19, 07: Determinants, cofactors, adjoints.

(29) Friday, April 20, 07: Matrix Inverse, using the inverse to solve linear systems.

Last two weeks of classes: Exam II and combinatorics (Chapter 6, Sections 7.1,7.2, 7.5-7.7, additional material).