Question

Let L be any language. Define even (w) as the string obtained by extracting from w the letters in even-numbered positions; that is, if

w = a₁a₂a₃a₄...,

then

even (w) = a₂a₄....

Corresponding to this, we can define a language

even (L) = {even (w) : w ∈ L}.

how can i solve this problem? help me please

Answer 1

here, w = a₁a₂a₃a₄...

even(w) = a₂a₄....

here even(w) is a part of w only, so it can also be written as :

even(w) = {w | letters in the even numbered positions of w}

Now, lets consider the regular expression:

r = (a*b*)*

so L(r) = {Ԑ, a, b, aa, ab, ba, bb,…, aaaa, aaab, aabb, abbb, abaa, abab, baaa, bbaa, bbba,…}

Considering L is a regular language and lets assume

if w = ‘abaa’

then even(w) = ’ba’ € L

Therefore even(L) is regular.