Question in computer scienceee

$*4.$ Rachmaninoff had big hands. Sergei Vasilyevich Rachmaninoff, the famous pianist, isknown to have big hands. As a practice for himself, he chose $56$ keys from a l$08-$key grandpiano 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 chosennumbers 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 applythe pigeonhole principle. To disprove a statement, construct a set S not satisfying thestatement as a counter$-$example. You have to figure out which statement$($s$)$ are true orfalse. You might need to apply pigeonhole principle more than once.$]$