Show that language is context free

Question

Show that language is context free
$E=\{a^ib^j|i\neq j\wedge 2i\neq j\}$

Look at my solution please: I use the fact that languages context free are closer under sum
$E=E_1\cup E_2\cup E_3 = \{a^ib^j|i>j\}\cup \{a^ib^j|j>2i\}\cup \{a^ib^j|i<j<2i\} $
Grammar for $E_1:$
$T\rightarrow a|aTb|aT$
Grammar for $E_2:$
$T\rightarrow aTbb|Tb|b$
Grammar for $E_3:$
$T\rightarrow aaU$
$U\rightarrow aUbV|bbb$
$V\rightarrow v|\epsilon$

For $E_1,E_2,E_3$ I showed CFG. For $E_1,E_2$ is obvious. What about grammar for $E_3$. Should I prove that it is correct and full ? Maybe, it is clear enough.

Answer 1

Assuming that you meant $V\to b\mid\epsilon$, it looks fine. As a teacher I think that I’d have wanted some justification for the grammar $E_3$, though. It would be enough to observe that if $U\to aUbV$ is applied $n$ times, the final result must be $a^{n+2}b^{n+3+k}$, where $k$ is the number of times that $V\to b$ is applied; clearly $0\le k\le n$, so

$$n+2<n+3+k\le 2n+3<2(n+2)\;.$$