# Section 2.4: Problem 6 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) Show that if
, then
.

(b) Show that it is not in general true that
.

(a) If
then, according to the Generalization Theorem,
, and, according to the axiom group 3,
, and, by MP,
.

(b) The problem here is that what (a) says is that for
and
such that
is valid,
is valid as well (well, it does not say anything about validity, but it will follow). However, in (b),
says that for any
and
, and any structure
and
, if
then
. Therefore, in (a) the statement is restricted to only those
and
for which
is valid, while in (b) there is no such restriction.

As an example, consider
a valid formula, and
the formula
. Now, if
has more than one element, and
is such that
, then
. However, while
,
, and, hence,
. Therefore,
.