Computer Scienceee

$*4.$ Rachmaninoff had big hands. Sergei Vasilyevich Rachmaninoff, the famous pianist, is known to have big hands. As a practice for himself, he chose $56$ keys from a $108-$key grand piano arbitrarily. If he opens his hand to play a $10$th $($that is$,$ having $9$ keys in between$),$ can he always find two chosen keys to play? How about playing an $11$th$?$

Let's rephrase the problem in set$-$theoretic language. Let $S$ be a set of $56$ arbitrarily chosen numbers from $1$ to $108.$ Prove or disprove the following:

$($a$)$ There are two numbers in $S$ that differ by exactly $10.$

$($b$)$ There are two numbers in $S$ that differ by exactly $11.$

$[$Hint: To prove a statement, specify the two sets and a function between them to apply the pigeonhole principle. To disprove a statement, construct a set $S$ not satisfying the statement as a counter$-$example. You have to figure out which statement$($s$)$ are true or false. You might need to apply pigeonhole principle more than once.$]$