Game Development Reference
In-Depth Information
Disjunctions
Unions and intersections allow you to explore how different sets share elements.
Situations also arise in which you find sets that share no common elements, but
you still want to show that they constitute a set. The logical term that you apply
to such situations is
or
. Consider, for example, the set of all elements contained in
sets
D
and
E
:
D ¼f
1, 2, 3, 4
g
E ¼f
6, 7, 8, 9
g
Set
D
contains numbers that are less than 5 and greater than 0, whereas set
E
contains numbers that are greater than 5 and less than 10. If you want to create a
set in which you can account for both of these sets, you can start with a logical
expression that employs the
or
symbol
ð_Þ
. For example, if
a
expresses any
number of the two sets, then
fa j a
<
5
^ a j a
>
0
g_ a j a
>
5
^ a j a
<
10
g
This expression allows you to say that the set includes numbers less than 5 and
greater than 0
or
numbers greater than 5 and less than 10. It so happens, however,
you can create a set
F
that consists of a union of these two sets:
F ¼fa j a
<
5
^ a j a
>
0
g[ a j a
>
5
^ a j a
<
10
g
Venn Diagrams
A Venn diagram allows you to easily show subsets, intersections, unions, and
disjunctions. To create a Venn diagram, you employ rounded figures that you
overlap and shade to show relationships between them. Figure 3.1 illustrates Venn
diagrams that represent different types of relationships between sets. Table 3.1
provides a discussion of each of the diagrams.
Exponents
When you multiply a number by itself, you raise it to a power. A power is a
product in which all of the factors are the same. To indicate that you are raising a
number to a given power, you employ an exponent. An exponent tells you the
