(1) — Basic notation.
•
Express in correct mathematical notation the set of all integers which are perfect squares,
that is, squares of other integers.
{
n
∈
Z
∃
m
∈
Z
, n
=
m
2
}
or
{
m
2

m
∈
Z
}
Popular incorrect answers:
{
n
∈
Z

n
=
m
2
}
(what is
m
?);
{
n
∈
Z

n
2
}
(
n
2
is a number,
not a statement that can be true or false);
{∃
n
∈
Z
...
}
,
{∀
n
∈
Z
...
}
(that’s not how
we write sets);
{
n,m
∈
Z

n
=
m
2
}
(ambiguous notation—is this a set of
n
∈
Z
, or
of
m
∈
Z
?);
{
n
∈
Z
∀
m
∈
Z
, n
=
m
2
}
(this would be the set of numbers which are
simultaneously squares of all integers, there are no such beasts); ‘
{
n

n
∈
S
}
, where
S
is
the set of perfect squares’ (you can’t use
S
to deﬁne
S
), etc.
•
Explain the difference between
{
1
,
2
,
3
}
and
(1
,
2
,
3)
.
{
1
,
2
,
3
}
denotes the set consisting of the numbers
1
,
2
,
3
, while
(1
,
2
,
3)
denotes an
ordered
triple of numbers, an element of a Cartesian product.
•
Explain the difference between
∅
and
{∅}
.
∅
denotes the empty set, with no elements, while
{∅}
denotes the set with one element, that
element being the empty set. The ﬁrst set is empty, the second set is not empty.
•
Explain the difference between the statements
(
∀
x
∈
R
)(
∃
y
∈
R
)
x < y
and
(
∃
y
∈
R
)(
∀
x
∈
R
)
x < y
The ﬁrst statement asserts that for every real number there exists a larger real number; this
is a true statement. The second asserts that there exists a real number which is larger than
every real number; this is a false statement.
Remark: See
§
1 of the notes if anything about all this is unclear.
2