Abstract Algebra

Instructions

• Assignments must be written in LaTeX with the final product being a PDF file.
• Use the Upload link above to submit your file by 11:59 pm on the due date. Late submissions will not be accepted.
• If you'd like, you can use the following LaTeX Homework Template.
• In all assignments, give justifications for your answers. For example, if you claim that a certain relation is not an equivalence relation because it is not transitive then show that it is not transitive.
• Problems taken from Chapter 1 - Preliminaries

Problems

1. Problem 2
2. Problem 21: Also, describe geometrically the equivalence classes (not the mere definition but a simple description that a high school student could understand). What is the equivalence class of \((3, 4)\)?
3. Problem 22: Parts (c), (d), (e) only.
4. Problem 25
5. Define a relation on \(\mathbb{R}^2\) by asking that \((x_1, y_1) \sim (x_2, y_2)\) if \(3x_1 + y_1 = 3x_2 + y_2\). It is straightforward to verify that this defines an equivalence relation (you are not required to do so but it would be a good exercise). Describe geometrically the equivalence classes.