1. AUGUST 30TH: BASIC NOTATION, QUANTIFIERS
1
1. August 30th: Basic notation, quantiﬁers
1.1. Sets: basic notation.
In this class,
Sets
will simply be ‘collections of
elements’. Please realize that this is not a precise deﬁnition; for example, we are not
saying what an ‘element’ should be. Our goal is not to develop set theory, which is
a deep and subtle mathematical ﬁeld; it is simply to become familiar with several
standard operations and with a certain language, and practice the skill of using this
language properly and unambiguously. What we are dealing with is often called
naive
set theory, and for our purpose we can rely on the intuitive understanding of what a
‘collection’ is.
The ‘elements’ of our sets will be anything whatsoever: they do not need to be
numbers, or particular mathematical concepts. Other courses (for example, Introduc
tion to Analysis) deal mostly with sets
of real numbers.
In this class, we are taking
a more noncommittal standpoint.
Two sets are equal if and only if they contain the same elements.
Here is how one usually speciﬁes a set:
S
=
{
·········
(the type of elements we are talking about)

“such that”
(some property deﬁning the elements of
S
)
}
For example, we can write:
S
=
{
n
even integer

0
≤
n
≤
8
}
to mean:
S
is the set of all even integers between
0
and
8
, inclusive.
This sentence is
the English equivalent of the ‘formula’ written above.
The

in the middle of the notation may also be written as a colon, “:”, or explicitly
as “such that”. On the other hand, there is no leeway in the use of parentheses:
S
= (
n
even integer

0
≤
n
≤
8)
or
S
= [
n
even integer

0
≤
n
≤
8]
are simply not used in standard notation, and they will not be deemed acceptable in
this class. Doing away with the parentheses,
S
=
n
even integer

0
≤
n
≤
8
is even worse, and hopefully you see why: by the time you have read ‘
S
=
n
’ you
think that
S
and
n
are the same thing, while they are not supposed to be. This
type of attention to detail may seem excessive, but it is not; we must agree on the
basic orthography of mathematics, and adhere to it just as we adhere to the basic
orthography of English.
An alternative way to denote a set is by listing all its elements, again within
{
—
}
:
S
=
{
0
,
2
,
4
,
6
,
8
}
again denotes the set of even integers between 0 and 8, inclusive. Since two sets are
equal precisely when they have the same elements, we can write
{
n
even integer

0
≤
n
≤
8
}
=
{
0
,
2
,
4
,
6
,
8
}
.
This is a true (if mindboggingly simple) mathematical statement, that is, a
theorem.
The ‘order’ in which the elements are listed in a set is immaterial:
{
0
,
2
,
4
,
6
,
8
}
=
{
2
,
0
,
8
,
4
,
6
}