2.1 Set theory
13
2.1.1
Basic terminology
Deﬁnition of a set:
•
A set is a collection of objects (concrete or abstract), called elements,
that usually share some common attributes, but are not otherwise re-
stricted in any fashion.
•
The curly brackets
{
and
}
are used as delimiters when specifying the
content of a set. This may be achieved by either listing all the elements
of the set explicitly, as is
{
1
,
2
,
3
,
4
,
5
,
6
}
(2.1)
or by stating the common properties satisﬁed by its elements, as in
{
a
:
a
is a positive integer
≤
6
}
(2.2)
In the latter case, the notation “
a
:” should read “all
a
such that”.
•
To indicate that an object
a
is an element of a set
A
, we write
a
∈
A
; we
also say that
a
is a member of, or belongs to,
A
. If
a
is not an element
of
A
, we write
a
6∈
A
.
•
Two sets
A
and
B
are identical (or equal) if and only if (iﬀ) they have
the same elements, in which case we write
A
=
B
. If
A
and
B
are not
identical, we write
A
6
=
B
.
•
Example: Let
A
=
{
1
,
2
,...,
6
}
and
B
=
{
2
,
4
,
6
}
. Then
A
6
=
B
because
1
∈
A
while 1
6∈
B
.
c
±
2003 Benoˆ
ıt Champagne
Compiled January 12, 2012