Math 360, Fall 2018, Assignment 5

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Read:

1. Section 4.

Carefully define the following terms, then give one example and one non-example of each:

1. Group of units (of a monoid).
2. $\mathrm{Fun}(S,S)$.
3. $\mathrm{Sym}(S)$.
4. $M_n(\mathbb{R})$.
5. $GL_n(\mathbb{R})$.
6. Isometry (of $\mathbb{R}^n$).
7. $\mathrm{Iso}(A)$ (where $A$ is some subset of $\mathbb{R}^n$).
8. $\mathbb{Z}_n$.

Carefully state the following theorems (you do not need to prove them):

1. Formula for the inverse of a product.
2. Theorem concerning injectivity of isometries.
3. Theorem concerning surjectivity of isometries.

Solve the following problems:

1. Section 4, problems 1, 2, 3, 4, 5, 6, 10, 11, 12, and 13.
2. In class, we asserted that every isometry of $\mathbb{R}^n$ is invertible. Now show that the inverse of an isometry is also an isometry.
3. Suppose that $f:A\rightarrow B$ is bijective. Show that for any $C\subseteq A$, we have $f^{-1}[f[C]]=C$, and for any $D\subseteq B$ we have $f[f^{-1}[D]]=D$. (Recall that on a previous assignment we saw that neither of these claims is true for general functions.)
4. Suppose now that $A$ is a subset of $\mathbb{R}^n$, and $f\in\mathrm{Iso}(A)$. Show that $f^{-1}\in\mathrm{Iso}(A)$; thus, $\mathrm{Iso}(A)$ is a group.

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