Confirm the following statements from Theorems $3\mathrm{.}2\mathrm{.}1$ and

$3\mathrm{.}2\mathrm{.}2.$

$($a$)$ A $+($B $+$ C$)=($A $+$ B$)+$ C

$($b$)($AB$)$C $=$ A$($BC$)$

$($c$)($a$+$ b$)$C $=$ aC $+$ bC

$($d$)$ a$($B $-$C$)=$ aB $-$ aC