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$.$ ainL; for any xinL,$xa$ and $xb$ are in $L.$

b$.$ ainL; for any xinL,$bx$ and $xb$ are in $L.$

c$.$ ainL; for any xinL,$ax$ and $xb$ are in $L.$

d$.$ ainL; for any xinL,$xb,xa,$ and $bx$ are in $L.$

e$.$ ainL; for any xinL,$xb,ax,$ and $bx$ are in $L.$

f$.$ ainL; for any xinL,$xb$ and xba are in $L.$

Hint: The non$-$recursive definition of the language listed at point a may be given as: "the set of all strings beginning with $a."$