Page 87 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Operations on Sets 73
(Hint: Use the fact, discussed in this section, that the universal quantifier
distributes over conjunction.)
7. Show that ∃x(P(x) → Q(x)) is equivalent to ∀xP(x) →∃xQ(x).
∗
8. Show that (∀x ∈ AP(x)) ∧ (∀x ∈ BP(x)) is equivalent to ∀x ∈
(A ∪ B)P(x). (Hint: Start by writing out the meanings of the bounded
quantifiers in terms of unbounded quantifiers.)
9. Is ∀x(P(x) ∨ Q(x)) equivalent to ∀xP(x) ∨∀xQ(x)? Explain. (Hint: Try
assigning meanings to P(x) and Q(x).)
10. (a) Showthat∃x ∈ AP(x) ∨∃x ∈ BP(x)isequivalentto∃x ∈ (A ∪ B)
P(x).
(b) Is ∃x ∈ AP(x) ∧∃x ∈ BP(x) equivalent to ∃x ∈ (A ∩ B) P(x)?
Explain.
∗
11. Showthatthestatements A ⊆ B and A \ B =∅ areequivalentbywriting
each in logical symbols and then showing that the resulting formulas are
equivalent.
12. Let T (x, y) mean “x is a teacher of y.” What do the following statements
mean? Under what circumstances would each one be true? Are any of
them equivalent to each other?
(a) ∃!yT (x, y).
(b) ∃x∃!yT (x, y).
(c) ∃!x∃yT (x, y).
(d) ∃y∃!xT (x, y).
(e) ∃!x∃!yT (x, y).
(f) ∃x∃y[T (x, y) ∧¬∃u∃v(T (u, v) ∧ (u = x ∨ v = y))].

2.3. More Operations on Sets

Now that we know how to work with quantifiers, we are ready to discuss some
more advanced topics in set theory.
So far, the only way we have to define sets, other than listing their elements
one by one, is to use the elementhood test notation {x | P(x)}. Sometimes this
notation is modified by allowing the x before the vertical line to be replaced
with a more complex expression. For example, suppose we wanted to define S
to be the set of all perfect squares. Perhaps the easiest way to describe this set
2
is to say that it consists of all numbers of the form n , where n is a natural num-
2
ber. This is written S ={n | n ∈ N}. Note that, using our solution for the first
statement from Example 2.2.3, we could also define this set by writing S =
2
2
2
{x |∃n ∈ N(x = n )}. Thus, {n | n ∈ N}={x |∃n ∈ N(x = n )}, and there-
2
2
fore x ∈{n | n ∈ N} means the same thing as ∃n ∈ N(x = n ).