Hey I want to check my solutions for this problem:
At the TUV, $n$ vehicles are inspected. For $i = 1, ..., n$, let $A_i$ denote the event "the i-th vehicle receives the inspection sticker." Describe the following events using set-theoretic operations on the events $A_i$:
(a) At least one of the $n$ vehicles does not receive a sticker.
(b) No vehicle receives a sticker.
(c) Exactly one vehicle does not receive a sticker.
(d) At most one vehicle receives a sticker.
My solutions:
a) Let $B$: "$0$ auto receives the inspection sticker" $B=(A_1 \cap A_2 \cap...\cap A_n )^c= (A_1^c \cup A_2^c \cup...\cup A_n^c) $
So be $\Omega$= "set of all possible entries"
We have for a) "At least one of the $n$ vehicles does not receive a sticker"= $\Omega \backslash B$
b) Let $C$= "No vehicle receives a sticker"=$A_1^c \cup ... \cup A_n^c$
c) D="Exactly one vehicle does not receive a sticker"= $A_1^c\cap A_2 \cap ... \cap A_n \cup A_1\cap A_2^c \cap ... \cap A_n \cup ... \cup A_1\cap A_2 \cap ... \cup A_n^c$
d) E="Exactly one vehicle receive a sticker"= $A_1\cap A_2^c \cap ... \cap A_n^c \cup A_1^c\cap A_2 \cap ... \cap A_n^c \cup ... \cup A_1^c\cap A_2^c \cap ... \cup A_n$
" At most one vehicle receives a sticker"=$B \cup E$
Have I done any mistakes? Thanks for the help