# Section 1.4: Problem 1 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

Suppose that
is generated from a set
by the binary operation
and unary operation
. List all the members of
. How many members might
have?
?

Let the maximum possible number of members of
be
. The maximum number means we assume that
is freely generated from
by
.
,
. For
, the sequence of length
has
or
as the first element, so that
, and the members with possible sequences are
,
,
,
,
, and
. For
, to produce a new element we either apply
to a function in
, or apply
to 2 different elements or two functions in a sequence in
, so that we get
more elements,
overall:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. Similarly for
.