Consider a Student Proposing Deferred Acceptance $($SPDA$)$ al$-$ gorithm on an one$-$one matching instance $$S$,$C $$ where $|$S$|=|$C$|=$ n$.$ Prove or disprove the following:

$1.$ For any instance, at least one student always makes multiple proposals. $2.$ There is an instance where the number of proposals made is n$($n$+1).$

$2$

$3.$ For any instance, there is always one college that receives exactly one proposal.

$4.$ For an instance n students and n colleges, the maximum number of pro$-$ posals that can be made is n$($n $1)+1.$