# Question: let be an alphabet given two words w1 and...

###### Question details

Let Σ be an alphabet. Given two words *w*_{1} and
*w*_{2}, a *match* between
*w*_{1} and *w*_{2} is a word
*w* satisfyingand for some and . Given
two languages *L*_{1} and *L*_{2}, we
use *L*_{1}♥*L*_{2} to denote the set
of matches between *w*_{1} and
*w*_{2} for all *w*_{1} ∈
*L*_{1}*,w*_{2} ∈
*L*_{2}. Prove that if *L*_{1} and
*L*_{2} are regular languages, then so is
*L*_{1}♥*L*_{2}.