The main subject of this course is basic concepts in set theory. After introducing some fundamental operations for sets such as intersection, union and complement, we explain basic notions for maps between sets, such as injection, surjection, and bijection. Next we introduce binary relations on sets, especially the concept of equivalence relation and the associated quotient set. Finally, we introduce the equivalence of sets, and learn the notion of cardinality. Each lecture will be accompanied by a problem solving class. This course will be succeeded by “Introduction to Topology II” in the second quarter.
The notions of set and map are fundamental not only in mathematics but also in science, and are applicable to describe a wide variety of objects. On the other hand, these abstract notions are not easy to comprehend without suitable training. To that end, rigorous proofs will be provided for most propositions, lemmas and theorems.
Students are expected to
・Understand De Morgan’s law
・Be familiar with injectivity, surjectivity, and bijectivity of mappings
・Be able to determine the image and preimage of maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets
set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course accompanied by discussion sessions
Course schedule | Required learning | |
---|---|---|
Class 1 | examples of sets, union, intersection and subset, complement | Details will be provided during each class session |
Class 2 | discussion session | Details will be provided during each class session |
Class 3 | De Morgan's law, distributive law, mapping between sets | Details will be provided during each class session |
Class 4 | discussion session | Details will be provided during each class session |
Class 5 | the image and preimage of map, composition of maps, product set | Details will be provided during each class session |
Class 6 | discussion session | Details will be provided during each class session |
Class 7 | correspondence between sets, indexed set | Details will be provided during each class session |
Class 8 | discussion session | Details will be provided during each class session |
Class 9 | binary relation, equivalence relation, equivalence class, quotient set | Details will be provided during each class session |
Class 10 | discussion session | Details will be provided during each class session |
Class 11 | the cardinality of set, relation between cardinality, countable set | Details will be provided during each class session |
Class 12 | discussion session | Details will be provided during each class session |
Class 13 | cardinality of the continuum, uncountable set, cardinality of power set | Details will be provided during each class session |
Class 14 | discussion session | Details will be provided during each class session |
Class 15 | evaluation of progress | Details will be provided during each class session |
None required
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
final exam (about 70%), discussion session (about 30%)
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation