Find an inductive definition for each sea S of strings am bn

Find an inductive definition for each sea S of strings. {a^m b^n | m, n N}. {a^2n | n N} {b^2n+1 | n N}.

Solution

d) Induction: If x S, then ax, xb S

(clearly as there can be any number times a and any number of times of b ,so if x S then ax, xb S

if x = a^mbⁿ  then by changing m -> (m+1) we,get ax ,and by changing n -> (n+1) ,we get xb )

j) Induction Let x S. If x = ay for some y, then aax S else bbx S (as set consists of either even number of times a or odd number of times b ,if x starts with a,then aax will also belong to S ,otherwise (when there is only b) then bbx will belong to S