Section 2.2: Problem 5 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
Show that the formula
(where
is a one-place function symbol and
is a two-place predicate symbol) is valid.
Let
be a structure and
. Then,
iff
or
iff
or
or
. So, assume that
does not hold, i.e.
. Then
, implying that
iff
iff
iff
, i.e. if
then at least one of the other two conditions holds. So, we have that for every
and
,
.