Page 82 - HOW TO PROVE IT: A Structured Approach, Second Edition

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68 Quantificational Logic
Note that for the third statement in this example we could not have given
the simpler answer ∃xL( j, x), because this would mean that John likes at least
one person, not exactly one person. The phrase exactly one occurs so often in
mathematics that there is a special notation for it. We will write ∃!xP(x)to
represent the statement “There is exactly one value of x such that P(x) is true.”
It is sometimes also read “There is a unique x such that P(x).” For example, the
third statement in Example 2.2.2 could be written symbolically as ∃!xL( j, x).
In fact, we could think of this as just an abbreviation for the formula given in
Example 2.2.2 as the answer for statement 3. Similarly, in general we can think
of ∃!xP(x) as an abbreviation for the formula ∃x(P(x) ∧¬∃y(P(y) ∧ y = x)).
Recall that when we were discussing set theory, we sometimes found it
useful to write the truth set of P(x)as {x ∈ U | P(x)} rather than {x | P(x)},to
make sure it was clear what the universe of discourse was. Similarly, instead
of writing ∀xP(x) to indicate that P(x) is true for every value of x in some
universe U, we might write ∀x ∈ UP(x). This is read “For all x in U, P(x).”
Similarly, we can write ∃x ∈ UP(x) to say that there is at least one value of x
in the universe U such that P(x) is true. For example, the statement ∀x(x ≥ 0)
would be false if the universe of discourse were the real numbers, but true if
it were the natural numbers. We could avoid confusion when discussing this
statement by writing either ∀x ∈ R(x ≥ 0) or ∀x ∈ N(x ≥ 0), to make it clear
which we meant.
As before, we sometimes use this notation not to specify the universe of
discourse but to restrict attention to a subset of the universe. For example, if
our universe of discourse is the real numbers and we want to say that some
2
real number x has a square root, we could write ∃y(y = x). To say that every
2
positive real number has a square root, we would say ∀x ∈ R ∃y(y = x). We
+
could say that every positive real number has a negative square root by writing
2
∀x ∈ R ∃y ∈ R (y = x). In general, for any set A, the formula ∀x ∈ AP(x)
−
+
means that for every value of x in the set A, P(x) is true, and ∃x ∈ AP(x)
means that there is at least one value of x in the set A such that P(x) is true. The
quantifiers in these formulas are sometimes called bounded quantifiers, because
they place bounds on which values of x are to be considered. Occasionally
we may use variations on this notation to place other kinds of restrictions on
quantified variables. For example, the statement that every positive real number
2
has a negative square root could also be written ∀x > 0∃y < 0(y = x).
Formulas containing bounded quantifiers can also be thought of as abbre-
viations for more complicated formulas containing only normal, unbounded
quantifiers. To say that ∃x ∈ AP(x) means that there is some value of x that
is in A and that also makes P(x) come out true, and another way to write
this would be ∃x(x ∈ A ∧ P(x)). Similarly, you should convince yourself