Sets are a powerful construct used in many mathematical, logical, and computer science-related applications. They are relied on heavily within the fields of machine learning, linguistics, and computing theory. The basic concepts of set theory can be expressed using several formats of set notation.<\/p>\n
Many symbols in set notation are ubiquitous among fields of logical design. For example, most equality operators have inverse inequality operators to mirror the logic expressed. Very few non-mathematicians have complete working knowledge of set notation. However, anyone studying within the fields of math or computer science can benefit from having a basic<\/em> working knowledge of the most common symbols and meaning of set notation.<\/p>\n A set is simply a collection of unique elements. Sets are often formally notated as a comma-separated series of elements contained within two curly brackets. For example, When comparing sets, two are considered equal if and only if they contain the exact same elements. It is worth noting that sets can contain any type of element and are not restricted to numerical values only. In computer programming, for example, a set could contain Objects, lists, pointers, strings, or any other comparable language construct.<\/p>\n Sets are defined in several ways, some using the notational syntax of others, some more informal than others. Below are some of the most common types of set notation one is likely to run into out in the wild.<\/p>\n Verbose set notation, also called semantic notation, is the “wordy” way to define sets. This uses natural language to express the conditions, members, and range of a given set. For example, to define a set of all integers in the range 1-5, inclusive, one would use the following semantic<\/em> notation:<\/p>\n Semantic Notation<\/strong>: Let S be the set whose members are the first 5 positive integers inclusive.<\/p>\n Roster notation, also known as enumeration notation, defines a set by explicitly listing members between the curly brackets. This notation is regarded as the informal notation but commonly used. It can use an ellipsis to abbreviate large spans of membership in a more concise form.<\/p>\n Sets are often used to describe an infinitely large collection of members. In such cases, the ellipsis notation is used again to indicate directional<\/em> infiniteness. For example, infiniteness in membership can be notated as follows:<\/p>\n Set-builder notation is similar to Roster and infinite notation. It can explicitly list membership, abbreviate membership using the ellipsis, and also represent unidirectional and bidirectional infiniteness. Set-builder notation is unique in that members are conditionally defined, commonly via means such as functions or predicates.<\/p>\n This set-builder notation represents the set V of all w (words) where the word starts with the letter a and is fewer than 5 characters in length. More simply; every word from the w section of the dictionary that’s fewer than 6 letters long.<\/p>\n Set-builder notation often utilizes a predicate to define members as well. This notation often makes more complex or cumbersome conditions expressible in more concise notation. For example, the set of all positive integers greater than 4 not in the set B can be expressed as such:<\/p>\nWhat’s in a Set, Anyway?<\/h2>\n
{1, 2, 3, 4, 5}<\/code> is a set<\/code> because all the elements are unique but {1, 1, 2, 3, 4, 5}<\/code> is not a set<\/code> because element 1 appears two times.<\/p>\nSet Notation<\/h2>\n
Verbose Notation<\/h3>\n
Roster Notation<\/h3>\n
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{1, 2, 3, 4, 5}<\/code><\/li>\n{1, 2, ..., 5}<\/code><\/li>\n<\/ul>\nInfinite Notation<\/h3>\n
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{1, 2, 3, ...}<\/code><\/li>\n{..., 1, 2, 3, ...}<\/code><\/li>\n<\/ul>\nSet-Builder Notation<\/h3>\n
W = {n | w starts with the letter 'a', and length(w) < 5}<\/code><\/p>\n