M13:  INTRODUCTION TO ABSTRACT MATHEMATICS

LECTURE:  M-W-F  9:00 -- 9:50 MSTB 124   (MAP:BLDG #415)
DISCUSSION:
Tu-Th  9:00 -- 9:50  MSTB 122   (MAP:BLDG #415)

INSTRUCTOR: Martin Zeman, RH 410E
OFFICE HOURS:  Wednesday, 10:00 - 12:00 (I may be few minutes late)   and by appointment

TA:  Thu Dinh, RH 425
OFFICE HOURS:   Tu,Th: 10:00-11:30

BOOK   By Alessandra Pantano and Neil Donaldson will be used as the text for this class.

HOMEWORK ASSIGNMENTS    HW1   HW2   HW3   HW4   HW5   HW6   HW7

COURSE PROGRESS       PREVIOUS WEEKS

WEEK 10
M: Proof of the proposition from Wednesday last week. If F:A-->B, the discussion when f^{-1}:B-->A. The relationship between dom(R) and dom(R^{-1} in general. Composition of functions and surjectivity, jectivity and bijectivity. Equinumerosity. Recommended practice problems: Book, Page 136 Exercise 7.4.1; Page 140 Exercise 7.4.2 and 7.4.3 and Page 144 Exercise 7.6.1 Please work through Section 7.4, 7.5, and 7.6 from the book.
W: Equinumerosity relation ~, partial ordering on the equivalence classes of ~, Schroeder-Bernstein theorem,injections, surjections, and passing between injections and surjections. Cardinality. Sets of finite cardinality and their properties. Recommended practice problems: Book, Page 151 Exercise 8.1.1, 8.1.6 and Page 152 Exercise 8.1.10 Please work through Section 8.1 from the book.

WEEK 9
M: Operations on quotients: Addition and multiplication on Z/divisibility mod n and correspondence with operations on remainders modulo n, construction of rational numbers as a quotient.
W: Inverse of a binary relation. Review of the domain, rangle, forward image and backward image of a set under a relation. Functions. Injective, surjective and bijective functions. Proposition: The inverse of a function f is a function iff f is injective.
F: Holiday.

WEEK 8
M: Examples of binary relations: The natural ordering relation on Z is a total ordering relation, the divisibility relation on N is an equivalence relation, the congruence modulo n is an equivalence relation on Z, inclusion is an ordering relation on the power set of A. Recommended practice problems: Book, Page 104 Exercise 6.2.1 -- 6.2.3 and Page 128 Exercise 7.3.1, 7.3.2, 7.3.6 Please work through Section 7.1 and 7.3 from the book.

W: Strict part of a partial ordering. Irreflexivity of a binary relation. Strict partial ordering: Irreflexivity and transitivity. Transition between a strict ordering relation and a non-strict one. Equivalence classes of an equivalence relation R on a set A. Quotient of a set by the equivalence relation R. Basic properties of equivalence classes of R:
- If y \in [x] then [y]=[x]
- (P1) Every x \in A is in some equivalence class of R.
- (P2) If [x],[y] are distinct then they are disjoint.
Recommended practice problems: Book, Page 128 Exercise 7.3.3 and Page 129 Exercise 7.3.10, 7.3.11
F: Equivalence relations. Equivalence classes. Quotient. Examples: Equality relation, divisibility modulo n.

WEEK 7
M: Holiday.
W: Completion of the proof of the Theorem from Friday for multiplication. Completion of the proof of well-ordering of N, quoting the result from the discussion about well-ordering of {1,...,n} for all n. Definition of an ordered pair.
F: Binary relations, Cartesian product, Binary relations from A to B and on A. Reflexivity, symmetricity, antisymmetricity, transitivity. Equivalence relations, partial and total ordering relations. Examples of relations: Equality relation on A is both an equivalence and partial ordering relation. Cartesian product Z x N. Recommended practice problems: Book, Page 99, Exercise 6.1.1 -- 6.1.3 and Page 119, Exercise 7.1.1 Please work through Section 6.1 and 6.2 from the book.