$2\mathrm{.}54$ Let G be the following grammar:

S $->$ Ta

T $->$ T aT b $|$ T bT a $|\backslash$epsi

a$.$ Show that L$($G$)=\{$wa$|$ w contains equal numbers of a$$s and b$$s$\}.$ Use a

proof by induction on the length of w$.$

b$.$ Use the DK$-$test to show that G is a DCFG$.$

c$.$ Describe a DPDA that recognizes L$($G$).$