Homework Assignment 01

Results to know

Please read over these problems. You can use these results in future problems even though you are not being asked to prove them.

• Section 1.1: 15, 19, 20, 22, 24, 28, 33, 34
• Section 1.2: 3, 7

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Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 1.1: 14, 25, 29, 31, 36
• Section 1.2: 1, 2, 5, 18

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment Homework 01 to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 1.1: 32
   • Consider starting like this: “Suppose $x^k = x^m$ for some $0\le k\le m< n$. Then…” With this approach, the goal is to show that $k=m$ (hence showing these powers of $x$ can only be equal if they are the same). You could also try a proof by contradiction. Whatever your approach, make sure to review the definition of the order of an element on page 20 of our book.
2. Section 1.1: 35
   • The Division Algorithm is a theorem; it is given in Section 0.2 of our book. You do not need to prove the parenthetical part of #35.
3. Section 1.2: 4
   • Please reread the six general properties of $D_{2n}$ listed on page 25 of our book. You are welcome to use them in your proofs.