And so to the final part of our investigation of language. We introduce the two quantifiers:
there exists, for all
In mathematics we frequently make existence assertions. For example the equationx2+ 2x + 1 = 0 has a real root
expresses an existence claim. The existential nature of this assertion can be made more explicit by rewriting it in the form:
there exists a real numberx such that x2+ 2x + 1 = 0 Or again
√2 is rational
also expresses an existence claim, though it certainly does not look much like an existence statement until we write it in the form
there exist natural numbersp and q such that√ 2= p/q
The first example here is a true statement (takea = −1); the second is false (as we prove later). Incidentally, the second example is not unique in having its existen-tial nature hidden: there are many mathematical statements that are really existence assertions, but that do not appear so on the surface.
We use the symbol
∃x to mean
there exists anx such that . . . Thus the above two examples may be written symbolically as:
• ∃x[x2+ 2x + 1 = 0]
• ∃p∃q[√
2= p/q]
But there is a potential problem with this notation. In the original English-language sentences we specified the kinds of objects involved in these quantifiers, real numbers in the first case and natural numbers in the second. But this is lost in the symbolic versions, where we rely upon the context or the reader’s experience (or both) to determine that thex is a real number variable and the p, q are variables for natural numbers.
Now, in point of fact, we often can (and do) simply rely on context, aided perhaps by the fairly loose conventions thatx and y usually denote real numbers, p, q, m, n denote whole numbers, andz, w denote complex numbers. (But do note that these conventions are very loose, and should not be relied upon without some additional indications of the usage.)
Another solution is to modify the quantifier notation so as to specify the kind of objects being quantified. For instance, we could make use of the symbolR for the set of real numbers (see Chapter 3) and write
(∃x ∈ R) to mean
there exists a real number such that. . .
(“x ∈ R” means “x is a member of R”. See Chapter 3 for a discussion of basic set theory.)
Thus, to say that the equationx2+ 2x + 1 = 0 has a real root we could write (∃x ∈ R)(x2+ 2x + 1 = 0)
Again, using the standard notationN for the set of natural numbers, we could write
(∃p ∈ N )(∃q ∈ N )[√
2= p/q]
to say that√
2 is rational.
To avoid proliferation of notation, where there is a repetition of a quantifier as here, we could (and often do) abbreviate this to
(∃p, q ∈ N )(√
2= p/q)
The back-to-front E that we use for the “exists” symbol comes from the word
“exists”.
As well as saying that certain objects exist, we also have frequent need to say that something holds for allx. We use the symbol
∀x to mean
for allx it is the case that . . .
Again, if we wish to specify what sort ofx we are considering, we can modify the notation, writing, for instance,
(∀x ∈ R) to mean
for all real numbersx it is the case that . . . and
(∀x ∈ N ) to mean
for all natural numbersx it is the case that . . . For example, to say that√
2 is irrational we can write (∀p ∈ N )(∀q ∈ N )(√
2 = p/q)
Notice that the symbol∀ is just an upside-down A, coming from the word “all”.
The symbol∃ is called the existential quantifier; ∀ is the universal quantifier.
Most statements in mathematics involve combinations of both kinds of quantifier.
For instance, to make the assertion that there is no largest natural number needs two quantifiers, thus:
(∀m ∈ N )(∃n ∈ N )(n > m)
This reads: for all natural numbersm it is the case that there exists a natural number n such that n is greater than m.
Notice that the order in which quantifiers appear can be of paramount importance.
For example, if we switch the order in the above we get (∃n ∈ N )(∀m ∈ N )(n > m)
This asserts that there is a natural number which exceeds all natural numbers — an assertion that is clearly false!
Exercises 2.3
(1) Express the following as existence assertions:
(a) The equationx3= 27 has a natural number solution.
(b) 1,000,000 is not the largest natural number.
(c) The natural numbern is not a prime.
(2) Express the following as “for all” assertions:
(a) The equationx3= 28 does not have a natural number solution.
(b) 0 is less than every natural number.
(c) The natural numbern is a prime.
(3) Express the following in symbolic form using quantifiers for people:
(a) Everybody loves somebody.
(b) Everyone is tall or everyone is short.
(c) Everone is tall or short.
(d) Nobody is at home.
(e) If a man comes, all the women will leave.
(4) Express the following using quantifiers. (In each case your quantifiers may refer only to the setsR and N .)
(a) The equationx2+ a = 0 has a real root for any real number a.
(b) The equationx2+ a = 0 has a real root for any negative real number a.
(c) Every real number is rational.
(d) There is an irrational number.
(e) There is no largest irrational number. (This one looks quite complex.) (5) LetC be the set of all cars, let D(x) mean that x is domestic, and let M(x)
mean thatx is badly made. Express the following in symbolic form using these symbols:
(a) All domestic cars are badly made.
(b) All foreign cars are badly made.
(c) All badly made cars are domestic.
(d) There is a domestic car that is not badly made.
(e) There is a foreign car that is badly made.
(6) Express the following sentence using quantifiers for real numbers:
There is a rational number between any two unequal real numbers.
(Use only logical connectives and quantifiers, the order relation<, and the symbolQ(x) having the meaning “x is rational”.)
(7) Express the following famous statement (by Abraham Lincoln) using quanti-fiers for people and times: “You can fool some of the people all of the time and all of the people some of the time, but you cannot fool all of the people all of the time.”