**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

**
COURSE INFORMATION AND POLICIES
UCI Academic Honesty Policy
UCI Student
Resources Page
**

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

**MIDTERM INSTRUCTIONS**
**MIDTERM PRACTICE PROBLEMS**

**FINAL EXAM PRACTICE PROBLEMS**

**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.**

**HOME**

Last Modified: December 6, 2019