Page 72 - HOW TO PROVE IT: A Structured Approach, Second Edition

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58 Quantificational Logic
the statement S(x) → (O(x) ∨ P(x)) can be false is if x is in that store, but
is neither overpriced nor poorly made. Thus, to say that the statement is
true for all values of x means that this never happens, which is exactly what
it means to say that everything in that store is either overpriced or poorly
made.
3. This means ¬(somebody is perfect), or in other words ¬∃xP(x), where
P(x) stands for “x is perfect.”
4. As in statement 2 in this example, we could think of this as meaning “If a
person dislikes Joe then Susan likes that person (no matter who the person
is).” Thus, we can start by rewriting the given statement as ∀x(if x dislikes
Joe then Susan likes x). Let L(x, y) stand for “x likes y.” In statements that
talk about specific elements of the universe of discourse it is sometimes
convenient to introduce letters to stand for those specific elements. In this
case we need to talk about Joe and Susan, so let’s let j stand for Joe and s for
Susan. Thus, we can write L(s, x) to mean “Susan likes x,” and ¬L(x, j) for
“x dislikes Joe.” Filling these in, we end up with the answer ∀x(¬L(x, j) →
L(s, x)). Notice that, once again, we have a universal quantifier applied to a
conditional statement. As before, you can check this answer using the truth
table for the conditional connective.
5. According to Definition 1.4.5, to say that A is a subset of B means that
everything in A is in B. If you’ve caught on to the pattern of how universal
quantifiers and conditionals are combined, you should recognize that this
would be written symbolically as ∀x(x ∈ A → x ∈ B).
6. As in the previous statement, we first write this as ∀x(x ∈ A ∩ B → x ∈
B \ C). Now using the definitions of intersection and difference, we can
expand this further to get ∀x[(x ∈ A ∧ x ∈ B) → (x ∈ B ∧ x /∈ C)].

Although all of our examples so far have contained only one quantifier,
there’s no reason why a statement can’t have more than one quantifier. For
example, consider the statement “Some students are married.” The word some
indicates that this statement should be written using an existential quantifier,
so we can think of it as having the form ∃x(x is a student and x is married).
Let S(x) stand for “x is a student.” We could similarly choose a letter to stand
for “x is married,” but perhaps a better analysis would be to recognize that to
be married means to be married to someone. Thus, if we let M(x, y) stand for
“x is married to y,” then we can write “x is married” as ∃yM(x, y). We can
therefore represent the entire statement by the formula ∃x(S(x) ∧∃yM(x, y)),
a formula containing two existential quantifiers.
As another example, let’s analyze the statement “All parents are married.”
We start by writing it as ∀x(if x is a parent then x is married). Parenthood,