5 Why is this important?

Set theory is important in computer science because many algorithms use this concept. In addition, set theory is the foundation on which more important theories are built, such as graph theory. Graph theory, though quite abstract, has practical applications in crytography as well as other areas of computer science.

It should be noted that some programming languages, such as Pascal, has built-in set types. Although the actual implementation of sets in Pascal is somewhat limited, it is still worthwhile to be mentioned. In C++, the STL (standard template library) also include a template class called set.

The implementation of sets depends on the application. A set is sparse if it contains a small number of elements out of a huge number of possible elements. The implementation of sparse sets often use some kind of data structures, such as linked lists. To speed up element look up (the $\in$ operator), a hash table may be used.

A set that is dense (containing most of the possible elements) can be implemented like a sparse one. However, for storage efficiency, dense sets can also be implemented by bitmaps. In a bitmap implementation, all elements are enumerated as integers. The enumeration position of an element indexes a bit position (in an huge array of bits) that reflects the presence of the element in a set. Set operations like union, intersection and difference are very efficient with a bitmap representation. However, the ``for ... in'' statement is less efficient compared to a hash table approach.