Mathematical Analysis I, 2004a

§§1–3. Sets and Operations on Sets. Quantifiers 5
where the vertical stroke stands for “such that.” For example, if N is again
the naturals, then the formula
(∀ x ∈ N | x>3) x ≥ 4 (1)
means “for each x ∈ N such that x>3, it is true that x ≥ 4.” In other words,
for naturals, x>3=⇒ x ≥ 4 (the arrow stands for “implies”). Thus (1) can
also be written as
(∀ x ∈ N) x>3=⇒ x ≥ 4.
In mathematics, we often have to form the negation of a formula that starts
with one or several quantifiers. It is noteworthy, then, that each universal
quantifier is replaced by an existential one (and vice versa), followed by the
negation of the subsequent part of the formula. For example, in calculus, a real
number p is called the limit of a sequence x 1 ,x 2 , ..., x n , ... iff the following
is true:
For every real ε>0, there is a natural k (depending on ε) such that, for
all natural n>k,wehave|x n − p| 0) (∃ k) (∀ n>k) |x n − p| 0)” and “(∀ n>k)” stand for “(∀ ε | ε>0)”
and “(∀ n | n>k)”, respectively (such self-explanatory abbreviations will also
be used in other similar cases).
Now, since (2) states that “for all ε>0” something (i.e., the rest of (2)) is
true, the negation of (2) starts with “there is an ε>0” (for which the rest of
the formula fails). Thus we start with “(∃ ε>0)”, and form the negation of
what follows, i.e., of
(∃ k) (∀ n>k) |x n − p| 0) (∀ k) (∃ n>k) |x n − p| ≥ε.
Note that here the choice of n>kmay depend on k. To stress it, we often
write n k for n. Thus the negation of (2) finally emerges as
(∃ ε>0) (∀ k) (∃ n k >k) |x nk − p| ≥ε. (3)
The order in which the quantifiers follow each other is essential. For example,
the formula
(∀ n ∈ N) (∃ m ∈ N) m>n