# Section 2.4: Problem 3 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

(a) Let
be a structure and let
. Define a truth assignment
on the set of prime formulas by

Show that for any formula (prime or not),

*Remark*: This result reflects the fact that and were treated in Chapter 2 the same way as in Chapter 1.

(b) Conclude that if
tautologically implies
, then
logically implies
.

(a) We show the statement by induction. For any prime formula
considered as a sentence symbol,
iff
. Now,
iff
iff (by induction)
iff
, and
iff
or
iff (by induction)
or
iff
. We conclude, that for every formula
,
iff
.

(b) If
tautologically implies
then for every structure
and
, if for every
,
, then
. Therefore, for every structure
and
, if for every
,
, then
. Hence,
.