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@@ -0,0 +1,105 @@ +\documentclass{article} +\usepackage[utf8]{inputenc} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amsmath} +\usepackage{enumerate} +\usepackage{hyperref} +\hypersetup{ + colorlinks, + citecolor=black, + filecolor=black, + linkcolor=black, + urlcolor=black +} + +% custom solution environment to set custom numbers +\theoremstyle{definition} +\newtheorem{innersolution}{Solution} +\newenvironment{solution}[1] +{\renewcommand\theinnersolution{#1}\innersolution} +{\endinnersolution} + +\title{Seminar exercises} +\author{ } +\date{February 2022} + +\begin{document} +\maketitle + +\begin{solution}{1.9}\ + \begin{enumerate}[1.] + \item Let $f(a) = u$, then $g(f(a)) = g(u)$, so $g \circ f$ is a function. + \item We can see that composition of functions is associative as follows:\\ + we know that $[ f \circ g](x) = f(g(x)), \forall x \in A$,\\ + so, + $$(h \circ [g \circ f])(x) = h([g \circ f](x)) = h(g(f(x)))$$ + \\ + and + $$([h \circ g] \circ f)(x) = [h \circ g](f(x)) = h(g(f(x)))$$ + Then, we can see that $$h \circ (g \circ f) = h(g(f(x))) = (h \circ g) \circ f$$ + \end{enumerate} +\end{solution} + +\begin{solution}{1.28}\ + + The quotient set of the equivalence relation in Example 1.27 is + $$ + X / \sim = \{[(x_0,y_0)], [(x_1, y_1)], \ldots, [(x_n, y_n)]\} + $$ + Yes, it is isomorphic to the cosets of the \emph{nth} roots of unity, which are $\mathbb{G}_n = \{w_k\}^{n-1}_{k=0}$, where $w_k=e^{\frac{2 \pi i}{n}}$. +\end{solution} + +\begin{solution}{2.2}\ + + To prove that the inverse $x^{-1}$ is unique, assume $x^{-1}$ and $\tilde{x}^{-1}$ are two inverses of $x$.\\ + By the definition of the inverse, we know that $x \cdot x^{-1} = e$. And by the definition of the unit element, we know that $x \cdot e = x$.\\ + Then, $$x^{-1} \cdot (x \cdot \tilde{x}^{-1}) = x^{-1} \cdot e = x^{-1}$$ + and $$(x^{-1} \cdot x) \cdot \tilde{x}^{-1} = e \cdot \tilde{x}^{-1} = \tilde{x}^{-1}$$ + By associativity property of groups, we know that + $$x^{-1} \cdot (x \cdot \tilde{x}^{-1}) = (x^{-1} \cdot x) \cdot \tilde{x}^{-1}$$ + so, $$x^{-1} \cdot e = e \cdot \tilde{x}^{-1}$$ + which is $$x^{-1} = \tilde{x}^{-1}$$ + So, for any $x \in G$, the inverse $x^{-1}$ is unique. +\end{solution} + +\begin{solution}{2.5}\ + + Let $\alpha = (\begin{smallmatrix}1 & 2 & 3\\ 1 & 3 & 2\end{smallmatrix})$, $\beta = (\begin{smallmatrix}1 & 2 & 3\\ 3 & 1 & 2\end{smallmatrix})$, then, + $$ + \alpha \cdot \beta = + (\begin{smallmatrix}1 & 2 & 3\\ 1 & 3 & 2\end{smallmatrix}) + \cdot (\begin{smallmatrix}1 & 2 & 3\\ 3 & 1 & 2\end{smallmatrix}) + = (\begin{smallmatrix}1 & 2 & 3\\ 3 & 2 & 1\end{smallmatrix}) + $$ + + and + $$ + \beta \cdot \alpha = + (\begin{smallmatrix}1 & 2 & 3\\ 3 & 1 & 2\end{smallmatrix}) \cdot + (\begin{smallmatrix}1 & 2 & 3\\ 1 & 3 & 2\end{smallmatrix}) + = (\begin{smallmatrix}1 & 2 & 3\\ 2 & 1 & 3\end{smallmatrix}) + $$ + + So, we can see that + $$ + (\begin{smallmatrix}1 & 2 & 3\\ 3 & 2 & 1\end{smallmatrix}) + \neq + (\begin{smallmatrix}1 & 2 & 3\\ 2 & 1 & 3\end{smallmatrix}) + $$ + + so, $\alpha \cdot \beta \neq \beta \cdot \alpha$. +\end{solution} + +\begin{solution}{2.26}\ + + We want to prove that $f: G \rightarrow H$ is a \emph{monomorphism} iff $\ker f=\{e\}$.\\ + We know that $f$ is a \emph{monomorphism} (\emph{injective}) iff $\forall a, b \in G$, $f(a) = f(b) \Rightarrow a = b$.\\ + Let $a, b \in G$ such that $f(a)=f(b)$. Then + $$f(a) f(b)^{-1} = f(b) (f(b))^{-1} = e$$ + $$f(a) f(b^{-1}) = e$$ + $$f(ab^{-1}) = e$$ + as $\ker f = \{e\}$, then we see that $ab^{-1}=e$, so $a=b$. Thus $f$ is a \emph{monomorphism}. +\end{solution} + +\end{document}