The question given is check Whether following is countable or not
$1).$The set of all the functions from $\mathbb{Q}$ to $\left \{0,1 \right \}$
$2).$The set of all the functions from $\mathbb{Q}$ to $\left \{0,1 \right \}$ which vanish outside a finite set
solution i tried- In $1$ option we can see that number of functions is $2^{\aleph_0}$ which is uncountable
But for $2$ option i am not getting clue how to proceed further Seems like he is asking about the cardinality of set of compact support functions .
Please help
Thank you