4 (a) Let L be an arbitrary languageover the alphabet {a, b}.
Define L* recursively. [10 Points]
Stet\Clrrn"e
i J\ e f
(!^a ! Fdz
J^/^t^t-<
x^,s,g
q,,^{ %,
re
f
4
*.@V
rt^ xyeF."
V-L
% V
Eftu^4CIat,qB
: trA'trwX;
& L+ ,rrfltut;^&t*,u4
{,1V^&"hZ*l )w,rbu&,p
c!^r*e"
c/
^1t",,\
(b)Usingyour definition of Lx prove by generalinduction(a.k.a.
(Lr f) Lz)* c Lt* O Lr*. [l0Points]
structuralinduction)that
Jt (t,A,l,)*, p^c4A€lf
finary : frr'fl,4+60h
{
,",,4;--eLfl A e L,\ n Lj
nL
: */f* /* ^rf,.*, n *(,n L$ ua"Lf
I^d^^fl+T<&
x4r-el^rwfihC*
* /,r"/,t Wrletn\n
f,T
I
^Hryu)NLf n/*Lj-'
f;ncs,
u e Lf nl)",
^*il
*-J,,\nL,.
un/,
xr^4u
0
/t'^l tr1eL!