Exercise $9($Ex$.44,$ Chapter $1$ of $[$Martin; $2011]).$

Each case below gives a recursive definition of a subset L of $\{$a$,$ b$\}$

$*$

$.$ Give a simple non$-$recursive

definition of L in each case.

a$.$ a in L; for any x in L$,$ xa and xb are in L$.$

b$.$ a in L; for any x in L$,$ bx and xb are in L$.$

c$.$ a in L; for any x in L$,$ ax and xb are in L$.$

d$.$ a in L; for any x in L$,$ xb$,$ xa$,$ and bx are in L$.$

e$.$ a in L; for any x in L$,$ xb$,$ ax$,$ and bx are in L$.$

f$.$ a in L; for any x in L$,$ xb and xba are in L$.$