Let P(x) be a property of an element x of a set E.
- ∀x∈E, P(x) (“for all x in E, P(x)”) is true when every element of E satisfies P;
- ∃x∈E, P(x) (“there exists x in E such that P(x)”) is true when at least one element of E satisfies P.
One writes ∃! for “there exists a unique”.
Examples
Example 1.6 (Negating everyday mathematical sentences)
Let f:R→R. The sentence “f is increasing” reads
∀x∈R, ∀y∈R,x≤y⟹f(x)≤f(y),
and its negation, by Proposition 1.5 plus the rule ¬(P⟹Q)⟺P∧¬Q:
∃x∈R, ∃y∈R,x≤y and f(x)>f(y):
a single witnessing pair suffices. Likewise “f is bounded” is ∃M∈R, ∀x∈R, ∣f(x)∣≤M, with negation
∀M∈R, ∃x∈R,∣f(x)∣>M:
whatever bound is proposed, some point beats it. The insight: a correct negation never contains “not” applied to a quantified block — it is a new positive statement, in which the roles are exchanged: one now produces the witnesses one previously received.
Example 1.7 (Order of quantifiers)
The order of different quantifiers matters:
∀x∈R, ∃y∈R, y>xis true (take y=x+1),
∃y∈R, ∀x∈R, y>xis false (no real number exceeds all reals).
In the first statement y may depend on x; in the second, one single y must work for all x. Two identical quantifiers, on the other hand, always commute.
Example 1.8 (Reading a definition with three quantifiers)
The sentence “the sequence (un) converges to ℓ” will be written in Chapter 11 as
∀ε>0, ∃N∈N, ∀n≥N,∣un−ℓ∣≤ε.
Its negation, by Proposition 1.5 applied three times, is
∃ε>0, ∀N∈N, ∃n≥N,∣un−ℓ∣>ε.
Being able to negate such sentences mechanically, without thinking about what they mean, is a genuine skill: it separates the logical work from the mathematical work.
Coach