For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z$,define $a ^* b = a - b$.
On $Z$,$^*$ is defined by $a ^* b = a - b$.
It can be observed that $1 ^* 2 = 1 - 2 = -1$ and $2 ^* 1 = 2 - 1 = 1$.
$\therefore 1 ^* 2 \neq 2 ^* 1$,where $1, 2 \in Z$.
Hence,the operation $^*$ is not commutative.
Also,we have:
$(1 ^* 2) ^* 3 = (1 - 2) ^* 3 = -1 ^* 3 = -1 - 3 = -4$.
$1 ^* (2 ^* 3) = 1 ^* (2 - 3) = 1 ^* (-1) = 1 - (-1) = 2$.
$\therefore (1 ^* 2) ^* 3 \neq 1 ^* (2 ^* 3)$,where $1, 2, 3 \in Z$.
Hence,the operation $^*$ is not associative.
It can be observed that $1 ^* 2 = 1 - 2 = -1$ and $2 ^* 1 = 2 - 1 = 1$.
$\therefore 1 ^* 2 \neq 2 ^* 1$,where $1, 2 \in Z$.
Hence,the operation $^*$ is not commutative.
Also,we have:
$(1 ^* 2) ^* 3 = (1 - 2) ^* 3 = -1 ^* 3 = -1 - 3 = -4$.
$1 ^* (2 ^* 3) = 1 ^* (2 - 3) = 1 ^* (-1) = 1 - (-1) = 2$.
$\therefore (1 ^* 2) ^* 3 \neq 1 ^* (2 ^* 3)$,where $1, 2, 3 \in Z$.
Hence,the operation $^*$ is not associative.
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