The sentence ``*x* + 2 = 5'' cannot be assigned a truth value and so it is
technically not a statement. However, if we knew the value of *x* we could
determine its truth value. We say that it is a statement involving a
*variable*,
namely *x*. If we allow *x* to have value 3 then we get a true statement
whereas, if we let *x* have value 1 we get a false statement. Symbolically,
we might represent the above statement as *p*(*x*). We see that *p*(3) is true
while *p*(1) is false. Similarly, the statement
*xy* = 4 could be represented symbolically as *q*(*x*,*y*) since it is a statement
involving two variables. Again, to assign a truth value to *q*(*x*,*y*) we would
have to know the values of both *x* and *y*. If *x* had value 2 and *y* had
value 5 then *q*(2,5) would have truth value F. Note that *q*(2,2) has truth
value T.

Another way to be able to assign truth values to statements involving
variables is through the use of *quantifiers*. The
*universal quantifier*,
, in English means ``For all possible values of the variable''.
The *existential quantifier*, , in English means
``There is a value of the variable''. Thus, with the examples above,
the following are statements:
; . The first states that for all
possible values of the variable *x* that *x* + 2 = 5. The second states that
there is a value of *x* for which *x* + 2 = 5. Clearly, we have made
each of these a statement (and we might believe the first to be false
and the second to be true). The problem we are left with is what do we
mean by the possible values of the variable? We assume that the variable in
our statement may be replaced by a quantity from a set of known objects.
This set of objects is called the *universal set of discourse*,
or more briefly, the *universal set*. Usually, this universal set will
be understood from the context. However, if necessary, we shall define the
universal set for each
variable in the statement and write something like .
Here, the notation translates into English as ``*x* an element
of the set *U*''. Note that for quantified statements involving more than
one variable, the universal set for each variable may be different.

Alternatively, we may think about a statement involving variables as a
statement about the universal set *U*. The *truth set* of the statement
consists of those elements of *U* for which the statement is true. For
example, if the universal set is the set of integers then the truth set of
the statement `` '' is . A statement about *U* is a
tautology if and only if its truth set is *U* and a contradiction if its
truth set is the *empty set*.
Thus a statement such as is true if its truth set is
*U*. If its truth set is a proper subset of *U* then it is a false statement.
A statement of the form is true if its truth set is
non empty and false if its truth set is the empty set.

Mon Sep 2 15:51:33 PDT 1996