2
1.
Sets (*)
A
set
is any collection of elements.
Examples:
a.
}
10
,
8
,
6
,
4
,
2
,
0
{
=
A
 the set of even numbers between zero and 10.
b.
}
,
,
{
bule
white
red
B
=
 the set of colors on the national flag.
c.
}
2
.
3
,

{
≥
=
GPA
feamale
students
M
of
U
C
 the set of U of M students that satisfy
the conditions listed after the vertical bar.
d.
}

)
,
{(
2
y
x
R
y
x
D
=
∈
=
 the set of vectors in the two dimensional Euclidean space,
such that the xcoordinate is equal to the ycoordinate.
e.
}
,
1
0
,
1
0

)
,
{(
2
y
x
y
x
R
y
x
E
≥
≤
≤
≤
≤
∈
=
 the set of vectors in the two
dimensional Euclidean space such that both coordinates are between 0 and 1 and the
xcoordinate is greater or equal to the ycoordinate. It is useful to illustrate this set
graphically.
The set E is colored blue.
Cartesian product
of A and B is sets the set of all ordered pairs such that the first
element belongs to A and the second belongs to B. We denote the Cartesian product by
B
A
×
.
Example:
}
3
,
2
,
1
{
=
A
,
}
8
,
7
{
=
B
, then
)}
8
,
3
(
),
7
,
3
(
),
8
,
2
(
),
7
,
2
(
),
8
,
1
(
),
7
,
1
{(
=
×
B
A
.
Convex sets:
A set B is convex if
]
1
,
0
[
)
1
(
,
∈
∀
∈
−
+
∈
∀
α
B
y
x
B
y
x
. In words, a
linear combination of any two elements in the set also belongs to the set.
0
0.2
0.4
0.6
0.8
1
1.2
0
0.1
0.3
0.5
0.7
0.9
1
x
y
E