Math 117 – April 05, 2011
3 Sets and Relations
A set
A
is a collection of objects characterized by a deﬁning property. If
A
is a set, then the sentence “
x
belongs to
A
” is a statement. We denote the property of the element
x
of belonging to the set
A
by
x
∈
A
.
A
is a
subset
of
B
, denoted by
A
⊆
B
, if (
x
∈
A
)
⇒
(
x
∈
B
). The sets are equal,
A
=
B
, if
A
⊆
B
and
B
⊆
A
. If
A
⊆
B
, and
B
*
A
, then
A
is called a
proper subset
of
B
, which is denoted by
A
(
B
.
The
empty set
,
∅
, is the set that doesn’t contain any element, i.e. the statement
x
∈
∅
is always
false. The empty set is a subset of every set.
Set Operations
:
A
∪
B
=
{
x
:
x
∈
A
or
x
∈
B
}
 union of
A
and
B
A
∩
B
=
{
x
:
x
∈
A
and
x
∈
B
}
 intesection of
A
and
B
A
\
B
=
{
x
:
x
∈
A
and
x /
∈
B
}
 compliment of
B
in
A
A universal set, denoted by
U
, is the set from which all the elements in a particular context originate.
For example in the context of functions of a real variable the universal set for the independent variable
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 Spring '08
 Akhmedov,A
 Set Theory, Sets, Empty set, Equivalence relation, Binary relation, Basic concepts in set theory, Set Operations

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