Question

Prove that the following languages are not regular using pumping lemma:

(a) {w : w != w^{R}}

(b) {a^{i} b^{j}a^{k} : k ≤ i + j}

Answer #1

Use the pumping lemma to show that the following languages are
not regular.
b. A2 = {www| w € {a, b}*}

Use the pumping lemma to show that {w | w belongs to {a,
b}*,and w is a palindrome of even length.} is not
regular.

Use pumping lemma to prove that L3a = {ab^m ab^m a| m>0} is
non-regular

Prove that regular languages are closed under the set dierence
operation. That is, if A and B are regular
languages, then, A - B is also a regular language.

Prove the Complement of Difference Lemma: ( A − B )' = A' ∪ B
using ONLY the set identities in the topical
notes.

{wRwwR | w ∈{a,b}∗}.
prove whether it's regular or not, if it is, draw a DFSM

Prove the following identity on languages A, B, C: A(B ∪
C) = AB ∪ AC
Find a counterexample to the following identity on
languages A, B: A* ∩ B* = (A∩B)*

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

The chopleft
operation of a regular language L is removing the leftmost symbol
of every string in L:
chopleft(L) = { w | vw
∈ L, with |v| = 1}.
Prove or disprove that
the family of regular languages is closed under the
chopleft operation.
Hint: If it’s regular,
give an idea of constructing an FA that accepts chopleft(L) using
an FA M that accepts L.
Otherwise, give a
counterexample.

Prove that the language A\B = {w: wx ∈ A, X ∈ B}, where A is a
CFL and B is regular is a CFL.

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