17: Everything You Wanted to Know about Sets in .NET framework

Encoder Denso QR Bar Code in .NET framework 17: Everything You Wanted to Know about Sets

Going for all (universal set) or nothing (empty set)

Consider the sets F = {Iowa, Ohio, Utah} and I = {Idaho, Illinois, Indiana, Iowa} Set F contains three elements, and set I contains four elements Within these sets, you can branch out to a set that you call the universal set for F and I You can also distinguish a set called the empty set, or null set This all or nothing business lays the foundation for performing set operations (see the section Operating on Sets later in this chapter) The following list presents the characteristics of these all or nothing sets: Universal set: A universal set for one or more sets contains all the possible elements in a particular category The writer of the situation must decide how many elements need to be considered in a particular problem But one characteristic is pretty standard: The universal set is denoted U For example, you could say that the universal set for F = {Iowa, Ohio, Utah} and I = {Idaho, Illinois, Indiana, Iowa} is U = {states in the United States} The universal set for F and I doesn t have to be a set containing all the states; it can just be all the states that start with a vowel Empty (or null) set: The opposite of the universal set is the empty set (or null set) The empty set doesn t contain anything (no kidding!) The two types of notation used to indicate the empty set are and { } The first notation resembles a zero with a slash through it, and the second is empty braces You must use one notation or the other, not both at the same time, to indicate that you have the empty set For example, if you want to list all the elements in set G, where G is the set that contains all the states starting with the letter Q, you write G = { } You have the empty set because no states in the United States start with the letter Q

The real world provides many special titles for the little guy Apartments can have sublets, movies can have subtitles, and ships can be submarines, so of course sets can have subsets A subset is a set completely contained within another set no element in a subset is absent from the set it s a subset of Whew! That s a mouthful, and my sentence ends with a preposition For the sake of good English, I ll start calling the sets subsets and supersets a superset representing what the subset is a subset of (there I go again!)

The set B = {2, 4, 8, 16, 32} is a subset of C = {x|x = 2n, n Z} Set B is a subset of set C, because B is completely contained in C Set C consists of all the numbers that are powers of 2, where the powers are all elements of the set of integers (Z) The notation for subset of is 1, and you write B 1 C to say that B is a subset of C The letter Z usually represents the integers, the positive and negative whole numbers, and zero Another way you can write the superset C is C = 'f, 1 , 1 , 1 , 1, 2, 4, 8, 16, 32, 64, 128, f1 8 4 2 using ellipses to indicate that the set continues on forever When one set is a subset of another, and the two sets aren t equal (meaning they don t contain the same elements), the subset is called a proper subset, indicating that the subset has fewer elements than the superset Technically, any set is its own subset, so you can say that a set is an improper subset of itself You write that statement with the subset notation and a line under it to indicate subset and, also, equal to To say that set B is its own subset, you write B 3 B This may seem like a silly thing to do, but, as with all mathematical rules, you have a good reason for doing so One of the reasons has to do with the number of subsets of any given set