Ways to Describe A Set

A set is a collection of unique objects.  A set can't contain two identical (indistinguishable) objects.  Consider the set of rational numbers, which are numbers that can be expressed as a/b, where a is an integer and b is a non-zero integer.  This set contains 2/4, and this set contains 4/8, because they can both be expressed as a/b.  But these are not two different elements of the set.  2/4 and 4/8 are two different ways of expressing the same element.  This means, for one thing, that if D is the set of all rationals except 2/4, then D does not contain 4/8.

Roster Notation

A list of elements, separated by commas, enclosed in curly braces.

Example: {3, 5, 7} is the set of single-digit odd prime numbers.

Tricky Example: { { }, {3}, {5}, {7}, {3,5}, {3,7}, {5,7}, {3,5,7} } is the set of subsets of the set of single-digit odd prime numbers.  Notice that every element of this set is itself a set.  The roster notation allows the use of nested curly-braces to describe sets which have other sets as elements.

Infinite set in roster notation: {1, 2, 3, ...} is the set of positive integers.  The first few elements illustrate the pattern, and the ellipsis (three dots) indicate that the pattern continues indefinitely.

Set-builder Notation

Set-builder notation is a shorthand way of saying "the set of all numbers, x, such that x has this property..."

Example: {a/b : a is an integer, b is a non-zero integer} is the set of rational numbers.

The colon means "such that".  Sometimes a vertical bar is used instead of a colon.

Interval Notation

An "interval" is a connected subset of the real number line.  Each end of an interval can be "open" or "closed".  An "open" end does not include its endpoint, and a "closed" end does.

Examples:

(2,3) means {x : x > 2 and x < 3}
[2,3) means {x : x ³ 2 and x < 3}
(2,3] means {x : x > 2 and x £ 3}
[2,3] means {x : x ³ 2 and x £ 3}

[2,¥) means {x : x ³ 2}
(-¥,3) means {x : x < 3}

Notice that if one end goes on to infinity (or negative infinity), this end must be an open end.  This is because infinity is not, itself, a real number, so it can't be included in any subset of real numbers.

Graphical Notation

A number line, representing the set of real numbers, is a thin line with an arrow at both ends.  The set of all numbers greater than a, where a is a real number, is represented by a darker line with a hollow point at a, and dark arrow to indicate that the set continues forever:

The set of all numbers greater than or equal to a is represented by a ray with a solid endpoint:

The set of numbers less than a is

The set of numbers less than or equal to a is

The set of numbers greater than or equal to a, and less then b is

Related pages in this website

Sets - how to construct sets of integers, reals, etc.

Functions - an introduction to the concept of a function, using set-builder notation to describe the "domain" and "range" of a function.

The webmaster and author of the Math Help site is Graeme McRae.
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