In the group $G=\{0, 1, 2, 3, 4, 5\}$ under addition modulo $6$,$(2 +_{6} 3^{-1} +_{6} 4)^{-1}$ is equal to

Determine which of the following binary operations on the set $N$ are associative and which are commutative. $a * b = \frac{a+b}{2}$ for all $a, b \in N$.

Consider the binary operations $^*: R \times R \rightarrow R$ and $o: R \times R \rightarrow R$ defined as $a \,^*\, b = |a-b|$ and $a \,o\, b = a$,$\forall \, a, b \in R$. Show that $^*$ is commutative but not associative,and $o$ is associative but not commutative. Further,show that $\forall \, a, b, c \in R, a \,^*\, (b \,o\, c) = (a \,^*\, b) \,o\, (a \,^*\, c)$. [If it is so,we say that the operation $^*$ distributes over the operation $o$]. Does $o$ distribute over $^*$? Justify your answer.

We define a binary relation $\sim$ on the set of all $3 \times 3$ real matrices as $A \sim B$ if and only if there exist invertible matrices $P$ and $Q$ such that $B = P A Q^{-1}$. The binary relation $\sim$ is

Show that $+: R \times R \rightarrow R$ and $\times: R \times R \rightarrow R$ are commutative binary operations,but $-: R \times R \rightarrow R$ and $\div: R_* \times R_* \rightarrow R_*$ are not commutative.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z^+$,define $a ^* b = a^b$.