The Use of Quantifiers
Open Statement: One that contains a variable, and becomes a
(single) statement when that variable is replaced with a value.
Example of an open statement:
7 divides evenly into x+7.
As you can see, for certain values of x, such as 7 or 21, this
statement is true, but for other values, it is not true. (Note: It is
possible for an open statement to always be true, such as “x is
greater than x1.”)
We can denote the open statement above as p(x). Thus we can
say that p(7) is true, whereas p(3) is not.
It is also possible for open statements to contain more than one
variable. Consider the following:
x is a prime number that divides into y evenly AND is less than
or equal to y.
We can denote the statement above as q(x,y). So, q(2,18) is true
while q(5,13) and q(27,3) are both false.
With both of these open statements, we see that there are
values for which the statements are true. Thus, it is reasonable
to say something like the following:
For some x, p(x), and
for some x and y, q(x,y).
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Quantifier Notation
Similarly, certain open statements can be true for all values of
the variables involved. If we let r(x) be the statement: “x is
greater than x1,” then we can make the claim:
For all x, r(x).
(Of course, whenever we say for all x, we must define what our
universe, or possible x’s are. Are they all the integers? All the
real numbers? All the complex numbers? We will talk about
how to specify that later.)
Since we are dealing with mathematics, we need short symbols
to replace the English words “for some” and “for all”. (A
mathematicians whole goal in life is to confuse non
mathematicians :))
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 Fall '09
 Logic, Prime number, Quantification, Universal quantification

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