Page 84 - HOW TO PROVE IT: A Structured Approach, Second Edition

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70 Quantificational Logic
no such value of x, simply because there isn’t a value of x for which x ∈ A is
true.
As an application of this principle, we note that the empty set is a subset
of every set. To see why, just rewrite the statement A ⊆ B in the equivalent
form ∀x ∈ A(x ∈ B). Now if A = ∅ then, as we have just observed, this
statement will be vacuously true. Thus, no matter what the set B is, ∅ ⊆ B.
Another example of a vacuously true statement is the statement “All unicorns
are purple.” We could represent this by the formula ∀x ∈ AP(x), where A is
the set of all unicorns and P(x) stands for “x is purple.” Since there are no
unicorns, A is the empty set, so the statement is vacuously true.
Perhaps you have noticed by now that, although in Chapter 1 we were al-
ways able to check equivalences involving logical connectives by making truth
tables, we have no such simple way of checking equivalences involving quan-
tifiers. So far, we have justified our equivalences involving quantifiers by just
looking at examples and using common sense. As the formulas we work with
get more complicated, this method will become unreliable and difficult to use.
Fortunately, in Chapter 3 we will develop better methods for reasoning about
statements involving quantifiers. To get more practice in thinking about quan-
tifiers, we will work out a few somewhat more complicated equivalences using
common sense. If you’re not completely convinced that these equivalences are
right, you’ll be able to check them more carefully when you get to Chapter 3.
Consider the statement “Everyone is bright-eyed and bushy-tailed.” If we let
E(x) mean “x is bright-eyed” and T (x) mean “x is bushy-tailed,” then we could
represent this statement by the formula ∀x(E(x) ∧ T (x)). Is this equivalent
to the formula ∀xE(x) ∧∀xT (x)? This latter formula means “Everyone is
bright-eyed, and also everyone is bushy-tailed,” and intuitively this means the
same thing as the original statement. Thus, it appears that ∀x(E(x) ∧ T (x)) is
equivalent to ∀xE(x) ∧∀xT (x). In other words, we could say that the universal
quantifier distributes over conjunction.
However, the corresponding distributive law doesn’t work for the existential
quantifier. Consider the formulas ∃x(E(x) ∧ T (x)) and ∃xE(x) ∧∃xT (x). The
first means that there is someone who is both bright-eyed and bushy-tailed, and
the second means that there is someone who is bright-eyed, and there is also
someone who is bushy-tailed. These don’t mean the same thing at all. In the
second statement the bright-eyed person and the bushy-tailed person don’t
have to be the same, but in the first statement they do. Another way to see the
difference between the two statements is to think about truth sets. Let A be the
truth set of E(x) and B the truth set of T (x). In other words, A is the set of
bright-eyed people, and B is the set of bushy-tailed people. Then the second
statement says that neither A nor B is the empty set, but the first says that A ∩ B
is not the empty set, or in other words that A and B are not disjoint.