set

A set is a group or collection of objects or numbers, considered as an entity unto itself. Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z. Each object or number in a set is called a member or element of the set. Examples include the set of all computers in the world, the set of all apples on a tree, and the set of all irrational numbers between 0 and 1.

When the elements of a set can be listed or denumerated, it is customary to enclose the list in curly brackets. Thus, for example, we might speak of the set (call it K) of all natural numbers between, and including, 5 and 10 as:

K = {5, 6, 7, 8, 9, 10}

A set can have any non-negative quantity of elements, ranging from none (the empty set or null set) to infinitely many. The number of elements in a set is called the cardinality, and can range from zero to denumerably infinite (for the sets of natural numbers, integers, or rational numbers) to non-denumerably infinite for the sets of irrational numbers, real numbers, imaginary numbers, or complex numbers).

The most basic relations in set theory can be summarized as follows.

• A set S₁ is a subset of set S if and only if every element of S₁ is also an element of S.
• A set S₁ is a proper subset of set S if and only if every element of S₁ is also an element of S, but there are some elements in S that are not elements of S₁.
• The intersection of two sets S and T is the set X of all elements x such that x is in S and x is in T.
• The union of two sets S and T is the set Y of all elements y such that y is in S or y is in T, or both.

Relationships between and among sets can be illustrated by means of a special type of drawing called a Venn diagram. The table below denotes common set symbology.

Set theory is fundamental to all of mathematics. In its "pure" form, set theory can be esoteric and even bizarre, and is primarily of interest to academics. However, set theory is closely connected with symbolic logic, and these fields are becoming increasingly relevant in software engineering, especially in the fields of artificial intelligence and communications security.

Read more about set: - Read more about "The Beginnings of Set Theory," by J.J. O'Connor and E.F. Robertson.

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