# Section2.2Exercises, Part I

##### 1

For each of the following, write Y if the given “operation” is a well-defined binary operation on the given set; otherwise, write N. In each case in which it isn't a well-defined binary operation on the set, provide a brief explanation. You do not need to prove or explain anything in the cases in which it is a binary operation.

1. $+$ on $\C^*$

2. $*$ on $\R^+$ defined by $x*y=\log_x y$

3. $*$ on $\M_2(\R)$ defined by $A*B=AB^{-1}$

4. $*$ on $\Q^*$ defined by $z*w=z/w$

##### 2

Define $*$ on $\Q$ by $p*q=pq+1\text{.}$ Prove or disprove that $*$ is (a) commutative; (b) associative.

##### 3

Prove that matrix multiplication is not commutative on $\M_2(\R)\text{.}$

##### 4

Prove or disprove each of the following statements.

1. The set $2\Z=\{2x\,:\,x\in \Z\}$ is closed under addition in $\Z\text{.}$

2. The set $S=\{1,2,3\}$ is closed under multiplication in $\R\text{.}$

3. The set

\begin{equation*} U=\left\{ \begin{bmatrix} a \amp b\\ 0 \amp c \end{bmatrix}\,:\,a,b,c\in \R\right\} \end{equation*}

is closed under multiplication in $\M_2(\R)\text{.}$ (Recall that $U$ is the set of upper-triangular matrices in $\M_2(\R)\text{.}$)

##### 5

Let $*$ be an associative and commutative binary operation on a set $S\text{.}$ An element $u\in S$ is said to be an idempotent in $S$ if $u*u=u\text{.}$ Let $H$ be the set of all idempotents in $S\text{.}$ Prove that $H$ is closed under $*\text{.}$