In the study of algebra, real numbers are often mentioned as a group, e.g. the set of real numbers greater than x, or the collection of real numbers satisfying a particular equation. Therefore, it is often convenient to indicate a particular group of real numbers using set notation. In this manner, the real numbers can be individually listed as part of a collection, or a continuum of real numbers can be represented concisely.
A set is a collection of objects. The objects are called the elements or members of the set. Conventionally, the pair of set braces, { } , are used to enclose the elements (or description thereof) of a set, using commas to separate the individual elements. Capital letters are normally used as names for sets.
If an object x is an element of set A, it is denoted by x Î A. Otherwise, the statement x Ï A, which means "x is not an element of A", is true.
The union of two sets A and B, denoted as A È B, is the set of all elements that are contained in sets A, B or both. The intersection of two sets A and B, denoted as A Ç B, is the set of all elements that belong to both sets A and B. An empty set has no elements; it is also called the null set and is denoted as Æ.
Two sets A and B are considered equal if they contain exactly the same elements. This is denoted by A = B. The statement A = B also implies that
A È B = A Ç B = A = B
Set-builder notation is generally used to represent a group of real numbers. It stipulates that sets be written in the format { x : x has property Y } , which is read as "the set of all elements x such that x has the property Y; the colon ":" means "such that". Using this notation, a set is often defined as the collection of real numbers that belong to either an open, closed, half-open, or infinite interval (of real numbers).
An open interval is a set of real numbers represented by a line segment of the real number line, whose endpoints are not included in the interval. This concept is made clear by the following definition:
( a, b ) = { x : a < x < b } where a < b
Since the endpoints of an open interval are not part of the interval,
a Ï ( a, b ) , b Ï ( a, b )
In contrast, a closed interval is a set of real numbers represented by a line segment of the real number line, whose endpoints are included in the interval. Its definition is as follows:
[ a, b ] = { x : a £ x £ b } where a < b
Since the endpoints of a closed interval are part of the interval,
a Î [ a, b], b Î [ a, b]
A half-open interval is also a set of real numbers represented by a line segment on the real number line, but with one endpoint included in the interval, and the other endpoint not included in the interval. They are defined as follows:
[ a, b ) = { x : a £ x < b } where a < b
( a, b ] = { x : a < x £ b } where a < b
An infinite interval is a set of real numbers, but it is represented by a ray or line on the real number line. As befits the name, an infinite interval does not have an endpoint in one or both directions. They are defined as follows:
(-¥, a ] = { x : x £ a }
[ a, -¥) = { x : x ³ a }
(-¥, ¥) = { x : x is a real number }
Similar definitions apply when the infinite interval is open at one end.
