Game Development Reference
In-Depth Information
Sets
A set is a collection of items. Mathematicians usually refer to the items as
ele-
ments
. A set is said to
contain
elements. The order of the elements makes no
difference. A set emerges from the
elements
that it contains. When you define a
set, such as the set of integers or counting numbers, you define the set of all
elements it can contain. You might not name all the elements, however, because a
set can consist of a finite or infinite number of elements. You still define the
condition by which you can determine whether a given item is or is not an
element of the set.
Elements
For example, you might picture a set of prime numbers. A prime number is a
positive integer that you can generate by multiplication using only the number
itself and 1. The other number must be distinct from the prime number. For this
reason, 1 is not a prime number.
Using such a definition, you can describe an infinite set of numbers. At the same
time, you can create sets of prime numbers using definitions that are more
restricted. Here is the set of prime numbers less than or equal to 47:
f
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
g
You identify a set using opening and closing curly braces, and you separate the
elements in a set using commas. To denote that a number is a member of a set,
you provide your set with a name. Although no strict rule applies, mathemati-
cians commonly represent the names of sets with italicized capital letters. You
might see the following, for example:
A ¼f
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
g
You can employ a small italic letter to represent the element symbolically. You
use
[
to designate that an element ''is a member of'' a set. If
a
equals 3, then
a 2 A
and 3
2 A
The number 1 is not a prime number, nor is 4. Assume that the value of
a
equals 1. To designate that 1 and 4 are not members of set
A
, you use the
following notation:
a
2 A
2 A
4
