Sets Overview
Mathematically, a set is a collection or list of objects.
Sets are not just comprised of numbers, but can contain anything including:
- the food in your refrigerator;
- the planets in the solar system;
Even though sets can contain anything, they often refer to numbers that fit a pattern or are related in some way such as the:
- set of positive even numbers less than 10: (0, 2, 4, 6, 8);
- set of factors for the number 12: (1, 2, 3, 4, 6, 12).
Set Notation
The objects in a set are called elements and the following notation or conventions are used with sets:
- Single uppercase letters are used to identify sets - such as J, E, or F;
- Lowercase letters or numbers are used for elements of a set;
- Curly braces { } denote a list of elements in a set;
- Commas are used to separate set elements.
So, examples of set notation would be:
J = {jupiter, saturn, uranus, neptune}
E = {0, 2, 4, 6, 8};
F = {1, 2, 3, 4, 6, 12};
Element Order and Repetition
Elements in a set do not have to be in any particular order so the set J above could also be written as:
J = {saturn, jupiter, neptune, uranus}
or
J = {neptune, jupiter, uranus, saturn}
Repeating elements does not change the set either, so:
J = {jupiter, saturn, uranus, neptune}
and
J = {jupiter, saturn, uranus, neptune, jupiter, saturn}
are the same set because both contain only four different elements: jupiter, saturn, uranus, and neptune.
Sets and Ellipses
If there is an infinite - or unlimited - number of elements in a set, an ellipsis (...) is used to show that the pattern of the set continues on forever in that direction.
For example, the set of natural numbers starts at zero, but has no end, so it can be written in the form:
{0, 1, 2, 3, 4, 5, …}
Another special set of numbers that has no end is the set of integers. Since integers can be positive or negative, however, the set uses ellipses at both ends to show that the set goes on forever in both directions:
{…, −3, −2, −1, 0, 1, 2, 3, …}
Another use for ellipses is to fill in the middle of a large set such as:
{0, 2, 4, 6, 8, …, 94, 96, 98, 100}
The ellipsis shows that the pattern - even numbers only - continues through the unwritten section of the set.
Special Sets
Special sets that are used frequently are identified using specific letters or symbols. These include:
- Ø or { } - the empty set - a set containing no elements;
- U - the universal set - a set containing all elements relative to a particular set definition;
- Z - the set of all integers: Z = {…, −3, −2, −1, 0, 1, 2, 3, …};
- N - natural numbers (positive integers): N = {0, 1, 2, 3, 4, 5, …}.
Roster vs. Descriptive Methods
Writing out or listing the elements of a set, such as the set of the inner or terrestrial planets in our solar system, is referred to as roster notation or the roster method.
T = {mercury, venus, earth, mars}
Another option for identifying the elements of a set is using the descriptive method, which uses a short statement or name to describe the set such as:
T = {the terrestrial planets}
Set-Builder Notation
An alternative to the roster and descriptive methods is to use set-builder notation, which is a shorthand method describing the rule that the elements of the set follow (the rule that makes them members of a particular set).
Set-builder notation for the set of natural numbers greater than zero is:
{x | x ∈ N, x > 0}
or
{x : x ∈ N, x > 0}
In set-builder notation, the letter "x" is a variable or placeholder, which can be replaced with any other letter.
Shorthand Characters
Shorthand characters that are used with set-builder notation include:
- The vertical bar or colon (| or : characters) - are separators read as such that;
- The lowercase epsilon (∈ character) - is read as is an element of;
- The ∉ character - is read as not an element of.
So, {x | x ∈ N, x > 0} would be read as:
"The set of all x, such that x is an element of the set of natural numbers and x is greater than 0."
Sets and Venn Diagrams
A Venn diagram - sometimes referred to as a set diagram - is used to show relationships between the elements of different sets.
In the image above, the overlapping section of the Venn diagram shows the intersection of sets E and F (elements common to both sets).
Below that is listed the set-builder notation for the operation (the upside down "U" means intersection):
E ∩ F = { x | x ∈ E , x ∈ F }
The rectangular border and the letter U in the corner of the Venn diagram represent the universal set of all elements under consideration for this operation:
U = { 0, 1, 2, 3, 4, 6, 8, 12}