# Section 2.2: Problem 7 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

Restate the definition of “
satisfies
with
” in the way described on page 84. That is, define by recursion a function
such that
satisfies
with
iff
.

For an atomic formula
(where
can also be the equality relation) we define
. Note, that for any atomic formula
,
iff
.

We further define
(i.e., by induction,
iff
iff
iff
),
(i.e., by induction,
iff
or
iff
or
iff
), and
(i.e., by induction,
iff for every
,
iff for every
,
iff
).