A set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set Theory. Here we give only a brief overview of the concept. |
A set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set theory. Here we give only a brief overview of the concept. |
Sets are one of the base concepts of mathematics. A set is, more or less, just a collection of objects, called its elements. Standard notation uses braces around the list of elements, as in:
{red, green, blue} {x : x is a primary color}
As you see, it is possible to describe one and the same set in different ways: either by listing all its elements (best for small finite sets) or by giving a defining property of all its elements.
If A and B are two sets and every x in A is also contained in B, then A is said to be a subset of B. Every set has as subsets itself, called the improper subset, and the empty set {}. The union of a collection of sets S = {S1, S2, S3, ...} is the set of all elements contained in at least one of the sets S1, S2, S3, ...} The intersection of a collection of sets T = {T1, T2, T3, ...} is the set of all elements contained in all of the sets. The union and intersection of sets, say A1, A2, A3, ... are denoted A1 u A2 u A3 u ... and A1 n A2 n A3 n ... respectively. If you don't mind jumping ahead a bit, the subsets of a given set form a boolean algebra under these operations. The set of all subsets of X is called its power set and is denoted 2X or P(X).
Examples of sets of numbers include
Statistical Theory is built on the base of Set Theory and Probability Theory.
For example: Suppose we call a set "well-behaved" if it doesn't contain itself as an element. Now consider the set S of all well-behaved sets. Is S well-behaved? There is no consistent answer; this is Russell's Paradox. In axiomatic set theory, no set can contain itself as an element.