Page 69 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/ P2: Oyk/
0521861241c02 CB996/Velleman October 19, 2005 23:48 0 521 86124 1 Char Count= 0

2

Quantificational Logic

2.1. Quantifiers

We have seen that a statement P(x) containing a free variable x may be true
for some values of x and false for others. Sometimes we want to say something
about how many values of x make P(x) come out true. In particular, we often
want to say either that P(x) is true for every value of x or that it is true for at least
one value of x. We therefore introduce two more symbols, called quantifiers,
to help us express these ideas.
To say that P(x) is true for every value of x in the universe of discourse
U, we will write ∀xP(x). This is read “For all x, P(x).” Think of the upside
down A as standing for the word all. The symbol ∀ is called the universal
quantifier, because the statement ∀xP(x) says that P(x)is universally true. As
we discussed in Section 1.3, to say that P(x) is true for every value of x in the
universe means that the truth set of P(x) will be the whole universe U. Thus,
you could also think of the statement ∀xP(x) as saying that the truth set of
P(x) is equal to U.
We write ∃xP(x) to say that there is at least one value of x in the universe
for which P(x) is true. This is read “There exists an x such that P(x).” The
backward E comes from the word exists and is called the existential quantifier.
Once again, you can interpret this statement as saying something about the
truth set of P(x). To say that P(x) is true for at least one value of x means that
there is at least one element in the truth set of P(x), or in other words, the truth
set is not equal to ∅.
For example, in Section 1.5 we discussed the statement “If x > 2 then
2
x > 4,” where x ranges over the set of all real numbers, and we claimed
that this statement was true for all values of x. We can now write this claim
2
symbolically as ∀x(x > 2 → x > 4).

55