# 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 $C$ is generated from a set $B=\{a,b\}$ by the binary operation $f$ and unary operation $g$ . List all the members of $C_{2}$ . How many members might $C_{3}$ have? $C_{4}$ ?
Let the maximum possible number of members of $C_{k}$ be $n_{k}$ . The maximum number means we assume that $C$ is freely generated from $B$ by $\{f,g\}$ . $n_{0}=0$ , $n_{1}=2$ . For $C_{2}$ , the sequence of length $2$ has $a$ or $b$ as the first element, so that $n_{2}=6$ , and the members with possible sequences are $C_{1}a$ , $C_{1}b$ , $ag(a)$ , $bg(b)$ , $af(a,a)$ , and $bf(b,b)$ . For $C_{3}$ , to produce a new element we either apply $g$ to a function in $C_{2}$ , or apply $f$ to 2 different elements or two functions in a sequence in $C_{2}$ , so that we get $18$ more elements, $24$ overall: $C_{2}a$ , $C_{2}b$ , $a\in C_{2}g(a)$ , $b\in C_{2}g(b)$ , $a\in C_{2}f(a,a)$ , $b\in C_{2}f(b,b)$ , $ag(a)g(g(a))$ , $bg(b)g(g(b))$ , $af(a,a)g(f(a,a))$ , $bf(b,b)g(f(b,b))$ , $[ab|ba]f(a,b)$ , $[ab|ba]f(b,a)$ , $ag(a)f(a,g(a))$ , $ag(a)f(g(a),a)$ , $ag(a)f(g(a),g(a))$ , $bg(b)f(b,g(b))$ , $bg(b)f(g(b),b)$ , $bg(b)f(g(b),g(b))$ , $af(a,a)f(a,f(a,a))$ , $af(a,a)f(f(a,a),a)$ , $af(a,a)f(f(a,a),f(a,a))$ , $bf(b,b)f(b,f(b,b))$ , $bf(b,b)f(f(b,b),b)$ , $bf(b,b)f(f(b,b),f(b,b))$ . Similarly for $n_{4}$ .