Chapter 6.2: Sets and Counting
•
A
set
is a welldefined collection of objects, called
elements
.
•
We can express sets by listing their elements, or by a definition.
Example:
A set expressed by listing:
A
=
{
1
,
2
,
3
}
A set expressed by a definition:
B
=
{
x

x
is even
}
•
Just like matrices, we can name sets with capital letters.
•
Sets may have a finite number of elements, like
A
, or an infinite number
of elements, like
B
.
We will focus on sets with a finite number of
elements.
•
We use the symbol
∈
to mean “is an element of”, and the symbol
/
∈
to mean “is not an element of”.
•
For example, we could write 3
∈
A
, because 3 is one of the elements
of
A
. We could also write 4
/
∈
A
because 4 is not one of the elements
of
A
.
•
We use the symbol
⊆
to show that one set is completely contained in
another. When we write something like
A
⊆
B
, we read it as “
A
is a
subset
of
B
”, and it means that everything inside of
A
is also inside
of
B
.
•
We use the symbol
notsubseteql
when we wish to say that one set is not com
pletely contained in another.
Example:
Let
A
=
{
1
,
5
,
7
}
,
B
=
{
1
,
3
,
5
,
7
,
10
}
, and
C
=
{
1
,
2
,
5
}
.
Since everything inside of
A
is also inside of
B
, we can write that
A
⊆
B
.
Since not everything inside of
C
is also inside of
B
, we can write
C
notsubseteql
B
.
•
We usually talk about sets in relation to a larger set, the
universal
set
, or
universe
.
The universe is a set which contains all the sets
under discussion. We use the symbol
U
for the universe.
•
When a set has no elements, we call it the
empty set
. We use the
symbol
∅
for the empty set.
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•
We can visualize sets by the use of a
Venn diagram
. With a Venn
diagram, we represent the universal set as a large rectangle, and the
other sets as circles within the universe.
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 Spring '11
 stephenlang
 Calculus, Set Theory, Sets, Counting, elements

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