The members of the set are often called the elements of the set. A set is usually named using
capital letters. Sets can
be defined in at least 3 ways:
1) Listing the members enclosed in brackets, either "{ }" or "(
)", with members of the set separated by commas.
2) Using a verbal description.
3) Using setbuilder notation.
An example of a set is the set of integers. That
was a verbal description for the set. We can also define the set of integers
using the other two ways to define sets:
Listing: I = {…, 3, 2, 1, 0, 1, 2, 3, ...}
SetBuilder Notation: I = {x  x is an integer}. This is read:
"I is the set of all x such that x is an integer."
We say two sets are equal when they have the same elements. For
example, if A is the set of primary colors and
B = {red, yellow, blue}, then A and B are equal. We could
write A = B. Another example might be if
W = {0, 1, 2, 3, ...} and T is the set of whole numbers, then we can say W = T. If the sets do not have the same elements,
they are not equal. For example, if W = {0, 1, 2, 3, ...} and
N = {1, 2, 3, ...}, then W is not equal to N since W has an element
that N does not have: the number 0. We can write W N.
When we list sets, we use two rules:
1) The same element is not listed more than once. So the set
A = {1, 2, 3, 3, 4, 5} should be written as A = {1, 2, 3,
4, 5}
2) The order of the elements does not matter. So the set
A = {1, 2, 3, 4, 5} = {1, 3, 5, 2, 4}.
We use the symbol to indicate that an object is a member
of a particular set. The symbol is used to indicate that an object is
not a member of a particular set. For example, if I represents the set of all
integers, then 100 I but 1/2 I.
The set with no elements is called the empty or null set. Both
of the symbols {} and are commonly used to denote this set.
For example, the set of all integers that are both negative and positive at the same time is
the empty set.
