OFFICE HOURS ZEMAN: M and W 12:00 - 1:30
in MSTB 261
OFFICE HOURS ANDREW-DAVID BJORK: M and Tue: 10:00 - 12:00 in RH 420B
LECTURE
(ZEMAN): M-W-F 9:00
- 9:50 MSTB 122 (MAP: BLDG #415)
DISCUSSION
(TBA): Tu - Th 9:00 - 9:50
MSTB 122 (MAP: BLDG
# 415)
NEXT QUIZ: No quizz anymore.
TOPICS
GRADED HOMEWORK: HW6 DUE: Friday, December 7
RULES
PREVIOUS
FINAL REVIEW SESSION: Tuesday, December
11 at 11:00 am in MSTB 114 RULES
WEEK
10
M: Equinumerosity, finiteness, countability. Equivalence relations
and ordering relations. Partitions induced
by equivalence relations.
5.15. Definition.
Equinumerosity.
5.16. Definition.
Finiteness, countability, uncountability.
6.1. Definition.
Reflexivity, symmetricity, antisymmetricity, transitivity.
6.2. Definition.
Equivalence relations, ordering relations.
6.3. Example.
(a) Equality on a set A.
(b) Congruence modulo k on Z (k
is positive)
(c) R on the Euclidean plane:
<a,b> \in R iff a,b have the same distance from the origin.
6.4. Definition.
Equivalence class of x with respect the equivalence relation R: [x]R,
the partition induced by R.
PROBLEMS FOR DISCUSSIONS:
Book, p.120: Exercise 26, 29, 31 and p.121: Exercise 34, 36,
37, 39
W: Partitions. Relationship between equivalence relations and partitions.
Quotient map.
6.5. Proposition.
A/R is really a partition, i.e. it splits A into disjoint nonempty sets.
Example:
Equivalence relation from 6.3(c).
6.6. Definition.
Quotient map kR: A --> A/R.
6.7. Proposition.
Quotient map is surjective.
6.8. Remark.
Quotient map is injective iff R is = .
6.9. Definition.
Partition. Equivalence relation induced by partition.
PROBLEMS FOR DISCUSSIONS:
Book, p.121 Exercise 42a, 43, 44.
F: Review