Show that every vector space has a basis using Zorn's lemma.
Show that for any positive integer $N$, the relation on $\mathbb{Z}$ given by $R_N=\{(a,b): a,b\in \mathbb{Z}, \text{ and } N \text{ divides } a-b\}$ is an equivalence relation.
Find an example of a relation which is:
(a) reflexive and symmetric but not transitive.
(b) symmetric and transitive but not reflexive.
(c) transitive and reflexive but not symmetric.
Let $f:X\to Y$ be a function. Define a relation $R$ of $X$ as follows: for $a,b\in X$, $aRb$ if $f(a)=f(b)$. Show that $R$ is an equivalence relation.
Let $\sim$ be an equivalence relation on a set $X$ and $Y=X/\sim$. Let $f:X\to Y$ be the function sending an element $a\in X$ to its equivalence class $[a]$. Let $R$ be the relation on $X$ associated to the function $f$ (as defined above). Show that $R=\sim$.
Let $(G,*,e)$ be a group containing exactly three elements. Show that $G$ is abelian.
Let $G$ be a group and $x,y\in G$ be such that $x\cdot y=y\cdot x$. Also assume that $<\{x\}>\cap<\{y\}>$ is the trivial subgroup. Show that $ord(xy)=lcm(ord(x),ord(y))$.
For $\sigma \in S_n$, show that $ord(\sigma)=lcm(n_1,\ldots,n_k)$ where $n_i=ord(\tau_i)$ and $\sigma=\tau_1\ldots\tau_k$ is a disjoint cycle decomposition of $\sigma$.
Let $G = \{ x \in \mathbb{R} : 0\le x<1\}$ and for $x, y \in G$ let $x * y$ be the fractional part of $x + y$ (i.e., $x * y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$). Prove that $*$ is a well defined binary operation on $G$ and that $G$ is an abelian group under * (called the real numbers mod 1).
Let $\mu_n=\{z\in \mathbb{C}: z^n=1\}$ for $n\in \mathbb{N}$ and $G=\cup_{n\ge 1}\mu_n$. Show that $\mu_n$ for all $n\in \mathbb{N}$ and $G$ are groups under multiplication. Also show that $\mu_3\cup \mu_5$ is not a subgroup of $G$.
Let $(\mathbb{Z}/n)^*=\{[a]\in \mathbb{Z}/n:(a,n)=1\}$. Show that this is a group under multiplication. Compute the order of all the element of this group for $n=12$.
Let $\sigma$ and $\tau$ be the following: $\sigma=\begin{pmatrix} 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ 3& 9& 7& 2& 11& 10& 6& 5& 1& 4& 8 \end{pmatrix}$ and $\tau=\begin{pmatrix} 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ 11& 10& 9& 8& 7& 6& 5& 3& 4& 1& 2 \end{pmatrix}$. Compute disjoint cycle decompositions of $\sigma$, $\sigma^2$ and $\sigma\tau$. Also compute their orders.