Lecture 2:
Sets, Relations, and Functions (a
Review)
Sets
•
A
set
is a collection of objects.
Example:
L
=
{
a, b, c, d
}
is a set of four
elements
.
•
a
∈
L
denotes
a
is
in
L
, and
z
∈
L
denotes
z
is not
in
L
.
•
The
empty set
, denoted by
∅
, contains no elements.
•
A set is
finite
if it has a finite number of elements.
Otherwise, the set is
infinite
.
•
A
is a
subset
of
B
, denoted
A
⊆
B
, if every element
in
A
is an element in
B
.
•
Two sets
A
and
B
are
equal
(
A
=
B
) if
A
⊆
B
and
B
⊆
A
.
•
A
⊂
B
means
A
is a
proper subset
of
B
(i.e.,
A
=
B
). By this definition,
∅
is a proper subset of any
nonempty set.
1
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Sets
Set operations.
Let
A
and
B
be sets.
•
Intersection
:
A
∩
B
=
{
x
:
x
∈
A
and
x
∈
B
}
. If
A
∩
B
=
∅
, we say that
A
and
B
are
disjoint
.
•
Union
:
A
∪
B
=
{
x
:
x
∈
A
or
x
∈
B
}
.
•
Difference
:
A
−
B
=
{
x
:
x
∈
A
and
x
∈
B
}
.
Power sets and partition.
•
The set of all subsets of a set
A
, denoted by 2
A
, is
called the
power set
of
A
.
For example, 2
{
c,d
}
=
{∅
,
{
c
}
,
{
d
}
,
{
c, d
}}
.
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 Spring '10
 Prof.Tai
 Empty set, Basic concepts in set theory

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