The importance of the membership problem

Given a word $w$ and a language $L$, we want to check if $w\in L$. This is called the membership problem. Why is the membership problem important?

The importance is that any computational problem where the answer is yes or no can be phrased as a membership problem in a language. The language is the set of strings for which the answer is yes.

StackExchange Computer Science Q#57154, answer score: 8