1)
Note: if we can able to give a CFG for a language L, then L is
context free
so,
CFG(Context free grammar): for given language:
S->A|B
A->aAc|D
D->bD|e //note here 'e' is empty string
B->cBa|D
since we have CFG for given language, it is Context free
1. Show that the following languae is context-free: {amb” cm:n, m >0} U {CPb9qP : P,...
Exercise 7.3.2: Consider the following two languages: Li = {a"b2ncm n,m >0} L2 = {a" mc2m | n,m >0} a) Show that each of these languages is context-free by giving grammars for each. ! b) Is L; n L, a CFL? Justify your answer.
Construct a context-free grammar for the language L={ab'ab'an> 1}.
Problem 8 You can assume that L = {a"be": n > 0} is not context free. Prove the following: Show that L-ab: n20 is not context free Show that L = {w E {a,b,c,d)* : na(w) = nb(w)-ne(w) = nd(w)) is not context free Note that na(w) means the number of a's in w .
Construct a context-free grammar for the language L={ ab"ab'an> 1}.
) Construct a context-free grammar for the language L={ ab”ab”a | n> > 1}.
Construct a context-free grammar for the language L={ ab”ab”a | n> 1}.
u(x,0)= Consider the following wave equation U, = U23 -00<x<00,t> 0 (0, -0<x<-1, _x+1, -1<< <0, 1-x, 0<x<1, 1<x<00 (0, -00<x<-1, u,(x,0) = 1, -15xs1, (0, 1<x<0. Find u(1,0.5) and u(-1,0.5).
1. Recursively define strings in the following language: A = {0"1"+mom nm >0} Then create a context-free grammar to describe the language.
(1) Suppose f :(M, d) + (N,0) is not uniformly continuous. Show that there exist an a > 0 and sequences (Xn) and (yn) in M such that d(Ion, yn) < and o(f(xn), f(n)) > € VnE N. (Hint: Negation of the definition of uniform continuity.)
2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1 = _(1 – p)p+1 – 1 p(n + 1)