Page 71 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Quantifiers 57
every case. In general, even if x is a free variable in some statement P(x), it is a
bound variable in the statements ∀xP(x) and ∃xP(x). For this reason, we say
that the quantifiers bind a variable. As in Section 1.3, this means that a variable
that is bound by a quantifier can always be replaced with a new variable without
changing the meaning of the statement, and it is often possible to paraphrase
the statement without mentioning the bound variable at all. For example, the
statement ∀xL(x, y) from Example 2.1.1 is equivalent to ∀wL(w, y), because
both mean the same thing as “Everyone likes y.” Words such as everyone,
someone, everything,or something are often used to express the meanings of
statements containing quantifiers. If you are translating an English statement
into symbols, these words will often tip you off that a quantifier will be needed.
As with the symbol ¬, we follow the convention that the expressions ∀x
and ∃x apply only to the statements that come immediately after them. For
example, ∀xP(x) → Q(x) means (∀xP(x)) → Q(x), not ∀x(P(x) → Q(x)).

Example 2.1.2. Analyze the logical forms of the following statements.

1. Someone didn’t do the homework.
2. Everything in that store is either overpriced or poorly made.
3. Nobody’s perfect.
4. Susan likes everyone who dislikes Joe.
5. A ⊆ B.
6. A ∩ B ⊆ B \ C.

Solutions
1. The word someone tips us off that we should use an existential quantifier.
As a first step, we write ∃x(x didn’t do the homework). Now if we let H(x)
stand for the statement “x did the homework,” then we can rewrite this as
∃x¬H(x).
2. Think of this statement as saying “If it’s in that store, then it’s either over-
priced or poorly made (no matter what it is).” Thus, we start by writing ∀x(if
x is in that store then x is either overpriced or poorly made). To write the
part in parentheses symbolically, we let S(x) stand for “x is in that store,”
O(x) for “x is overpriced,” and P(x) for “x is poorly made.” Then our final
answer is ∀x[S(x) → (O(x) ∨ P(x))].
Note that, like statement 4 in Example 2.1.1, this statement has the form
of a universal quantifier applied to a conditional statement. This form occurs
quite often, and it is important to learn to recognize what it means and when
it should be used. We can check our answer to this problem as we did before,
by using the truth table for the conditional connective. The only way that