Page 73 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Quantifiers 59
like marriage, is a relationship between two people; to be a parent means to
be a parent of someone. Thus, it might be best to represent the statement “x
is a parent” by the formula ∃yP(x, y), where P(x, y) means “x is a parent
of y.” If we again represent “x is married” by the formula ∃yM(x, y), then
our analysis of the original statement will be ∀x(∃yP(x, y) →∃yM(x, y)).
Although this isn’t wrong, the double use of the variable y could cause confu-
sion. Perhaps a better solution would be to replace the formula ∃yM(x, y) with
the equivalent formula ∃zM(x, z). (Recall that these are equivalent because a
bound variable in any statement can be replaced by another without changing
the meaning of the statement.) Our improved analysis of the statement would
then be ∀x(∃yP(x, y) →∃zM(x, z)).

Example 2.1.3. Analyze the logical forms of the following statements.

1. Everybody in the dorm has a roommate he doesn’t like.
2. Nobody likes a sore loser.
3. Anyone who has a friend who has the measles will have to be quarantined.
4. If anyone in the dorm has a friend who has the measles, then everyone in
the dorm will have to be quarantined.
5. If A ⊆ B, then A and C \ B are disjoint.

Solutions
1. This means ∀x(if x lives in the dorm then x has a roommate he doesn’t
like). To say that x has a roommate he doesn’t like, we could write ∃y(x
and y are roommates and x doesn’t like y). If we let R(x, y) stand for
“x and y are roommates” and L(x, y) for “x likes y,” then this becomes
∃y(R(x, y) ∧¬L(x, y)). Finally, if we let D(x) mean “x lives in the dorm,”
then the complete analysis of the original statement would be ∀x[D(x) →
∃y(R(x, y) ∧¬L(x, y))].
2. This is tricky, because the phrase a sore loser doesn’t refer to a particular
sore loser, it refers to all sore losers. The statement means that all sore losers
are disliked, or in other words ∀x(if x is a sore loser then nobody likes x). To
say nobody likes x we write ¬(somebody likes x), which means ¬∃yL(y, x),
where L(y, x) means “y likes x.” If we let S(x) mean “x is a sore loser,” then
the whole statement would be written ∀x(S(x) →¬∃yL(y, x)).
3. You have probably realized by now that it is usually easiest to translate from
English into symbols in several steps, translating only a little bit at a time.
Here are the steps we might use to translate this statement:
(i) ∀x(if x has a friend who has the measles then x will have to be quaran-
tined).