NTA UGC NET JUNE-2023 Paper-2

The language L = {w in {a, b, c}* : na(ω) + nb(ω) = nc(ω} is indeed context-free. It is not linear, but it is context-free.

Here’s an example of a context-free grammar that generates this language:

S -> aSc (This rule adds one ‘a’ and one ‘c’ to the string, maintaining the balance.)
S -> bSd (This rule adds one ‘b’ and one ‘d’ to the string, maintaining the balance.)
S -> ε (This rule allows the string to be empty.)
Using this context-free grammar, you can generate strings that satisfy the condition na + nb = nc. For example:

For na = 2 and nb = 2, you can generate aacbcd.
For na = 3 and nb = 3, you can generate aabbcc.
So, the language is context-free but not linear.