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 \(*\text{.}\)