$(10$ points$)$ Determine which of the following binary operations are as$-$ sociative. Please justify your work. If it is false provide a counterexam$-$ ple, if not then show your mathematical argument.

$$ thebinaryoperation$$onZdefinedbya$$b$=$a$+$b$+$ab

$$ the binary operation $$ on Q defined by a $$ b $=$ a$+$b

$5$

$$ thebinaryoperation$$onZdefinedbya$$b$=$a$+|$b$|$

$$ the binary operation $$ on Q $\backslash \{0\}$ defined by a $$ b $=$ ab

$$ the binary operation $$ defined on R $\backslash$times R by $($a$,$b$)($c$,$d$)=($ac $$ bd$,$ad$+$bc$)$