Page 81 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equivalences Involving Quantifiers 67
younger than x. In fact, this suggests that a good way of reading the pair of
quantifiers ∃y∃x or ∃x∃y would be “there are objects x and y such that. . . .”
Similarly, two universal quantifiers in a row can always be switched without
changing the meaning of a formula, because ∀x∀y and ∀y∀x can both be
thought of as meaning “for all objects x and y, . . . .” For example, consider
the formula ∀x∀y(L(x, y) → A(x, y)), where L(x, y) means “x likes y” and
A(x, y) means “x admires y.” You could think of this formula as saying “For
all people x and y,if x likes y then x admires y.” In other words, people always
admire the people they like. The formula ∀y∀x(L(x, y) → A(x, y)) means
exactly the same thing.
It is important to realize that when we talk about objects x and y,weare
not ruling out the possibility that x and y are the same object. For example,
the formula ∀x∀y(L(x, y) → A(x, y)) means not just that a person who likes
another person always admires that other person, but also that people who like
themselves also admire themselves. As another example, suppose we wanted
to write a formula that means “x is a bigamist.” (Of course, x will be a free
variable in this formula.) You might think you could express this with the
formula ∃y∃z(M(x, y) ∧ M(x, z)), where M(x, y) means “x is married to y.”
But to say that x is a bigamist you must say that there are two different people
to whom x is married, and this formula doesn’t say that y and z are different.
The right answer is ∃y∃z(M(x, y) ∧ M(x, z) ∧ y = z).

Example 2.2.2. Analyze the logical forms of the following statements.

1. All married couples have fights.
2. Everyone likes at least two people.
3. John likes exactly one person.

Solutions
1. ∀x∀y(M(x, y) → F(x, y)), where M(x, y) means “x and y are married to
each other” and F(x, y) means “x and y fight with each other.”
2. ∀x∃y∃z(L(x, y) ∧ L(x, z) ∧ y = z), where L(x, y) stands for “x likes y.”
Note that the statement means that everyone likes at least two different
people, so it would be incorrect to leave out the “y = z” at the end.
3. Let L(x, y) mean “x likes y,” and let j stand for John. We translate this
statement into symbols gradually:
(i) ∃x(John likes x and John doesn’t like anyone other than x).
(ii) ∃x(L( j, x) ∧¬∃y(John likes y and y = x)).
(iii) ∃x(L( j, x) ∧¬∃y(L( j, y) ∧ y = x)).