Writing Assignment 05

Directions.

1. Get the template for this assignment (or use your own). Here’s how to do it:
   • Go to www.overleaf.com, and make sure you are logged in.
   • In a new window, go here: https://www.overleaf.com/read/yttktyzqswhy
   • Click on the menu icon in the upper-left and select “Copy Project”.
   • Enter a name and click “Copy”.
   • When this completes you will be back in your own workspace (instead of mine).
2. Use $\LaTeX$ to type up your proofs of the Propositions to Prove listed below. Make sure to use complete sentences and appropriate punctuation. Also, make sure to edit for typos.
3. Click on the “Download PDF” button (to the right of the “Recompile” button). Save the file somewhere you can easily find it. Submit the pdf in Canvas.
4. Let me know if you have any questions!

Remember, if you have trouble finding the command for a math symbol you want to use, try googling it, email me, or try looking in this Short Math Guide for $\LaTeX{}$.

Propositions to Prove.

1. Consider $(\mathbb{R}^3,+)$, where $\mathbb{R}^3$ is the set of all 3-entry row vectors with real number entries and + is ordinary vector addition. Let $K$ be the subset of $\mathbb{R}^3$ consisting of vectors whose entries sum to 0; that is $K:={(a,b,c)\in \mathbb{R}^3\mid a+b+c = 0}$. Prove that $K$ is a subgroup of $(\mathbb{R}^3,+)$. (See Problem 3.15(b). You do not need to prove that $(\mathbb{R}^3,+)$ is a group.)

2. For any $n\in \mathbb{Z}$, define $n\mathbb{Z} := {nk\mid k\in \mathbb{Z}}$. Prove that $n\mathbb{Z}$ is a subgroup of $(\mathbb{Z},+)$. (See Problem 3.16(c).)