In this lesson, you will learn the logic necessary to describe propositions, ideas necessary to demonstrate inclusion relations
of sets, and distance spaces which are the Euclid spaces and its abstraction. Further abstraction (topological spaces) is
learned in "Geometry I" class.
Modern mathematics is based on the idea of "sets". For example, in mathematics up to high school, it was the main to investigate
the properties of individual sequences and functions given, but in university mathematics, conversely, to consider sequences
having the same property ("the set of such sequences"), and functions having the same property ("the set of such sequences"),
etc. By introducing the concept of inner product and distance into such a set, we will consider the abstraction (distance
spaces) of Euclid space by making something similar to the world we know well.
- You can describe propositions logically.
- You can demonstrate properties of sets and maps.
- You can demonstrate properties of sets in Euclidean spaces.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1. | Logic 1: propositional logic, truth table | Check out "Logic" in advance. | 190minutes |
| 2. | Logic 2: predicate logic, \forall, \exists | Review the previous session in advance. | 190minutes |
| 3. | Set 1: Fundamentals of sets | Review the previous session in advance. Check out "Set" in advance. | 190minutes |
| 4. | Set 2: family of sets, direct product | Review the previous session in advance. | 190minutes |
| 5. | Set 3: equivalence relation, equivalence class | Review the previous session in advance. | 190minutes |
| 6. | Map 1: inverse image, surjection, injection | Review the previous session in advance. Check out "Map" in advance. | 190minutes |
| 7. | Map 2: composition | Review the previous session in advance. | 190minutes |
| 8. | Attention on Mid-term assignment | Review Session2 1-7 in advance. | 190minutes |
| 9. | Cardinality: countable set, diagonal argument, power set, cardinality of the continuum | Review "surjection" and "injection". | 190minutes |
| 10. | 1-dim Euclidean space 1: \epsilon-neighbourhood, adherent point, accumulation point, inner point, exterior point, boundary point, closure, closed set | Check out "Geometry and Equations" in advance. | 190minutes |
| 11. | 1-dim Euclidean space 2: open set | Review the previous session in advance. | 190minutes |
| 12. | n-dim Euclidean space: \epsilon-neighbourhood, Heine-Borel's theorem | Review the previous session in advance. | 190minutes |
| 13. | Metric space: distance function, example | Review the previous session in advance. | 190minutes |
| 14. | Attention on Final assignment | Review Sessions 9-13 in advance. | 190minutes |
| Total. | - | - | 2660minutes |
| Mid-term assignment | Final assignment | Total. | |
|---|---|---|---|
| 1. | 30% | 10% | 40% |
| 2. | 20% | 10% | 30% |
| 3. | 30% | 30% | |
| Total. | 50% | 50% | - |
Mid-term assignment and Final assignment. As a criterion, if you determine that you understand 60% of the content covered
in the class, your final score will be 60 points.
Ken Kuriyama, "Ronri, Syugou to Isoukuukan nyumon", Kyoritu-shuppan.
Check out "Logic", "Set", "Map", and "Geometry and Equations" in your high-school textbook.
- Course that cultivates an ability for utilizing knowledge
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | N/A |
Last modified : Sun Mar 21 16:01:07 JST 2021