Which of the following are context free language?
A. {$$w^ix^jy^kz^l$$​ | i + k = j + l, where i, j, k, $$l\geq0$$​}
B. { $$w^ix^jy^kz^l$$​ | i = j and k = l, where i, j, k, $$l\geq0$$​}
C. { $$w^ix^jy^kz^l$$​ | i = j = k and $$k \neq l$$​, where i, j, k, $$l\geq0$$​}
D. { $$w^ix^jy^kz^l$$​ | i = j + l, where i, j, k, $$l\geq0$$​}
E. { $$w^ix^jy^kz^l$$​ | i = j = l and $$k \neq l$$​, where i, j, k, $$l\geq0$$​}
Choose the correct answer from the options given below:

Correct option is A

The correct answer is (a) A, B only because the conditions in statements A and B define constraints that can be represented with context-free languages (CFLs). Context-free languages can be recognized by pushdown automata (PDAs), which are powerful enough to handle certain patterns and equalities, but not arbitrary inequalities or conditions that require multiple levels of nesting.
Information Booster:
1. Context-Free Languages (CFLs) are languages that can be generated by context-free grammars and recognized by PDAs.
2. Statement A involves a sum condition between indices, which is achievable with a PDA by tracking the count difference between parts of the string.
3. Statement B has a simple equality condition (i = j and k = l), which a PDA can handle by balancing counts in a nested structure.
Additional Knowledge:
· Statement C involves a triple equality and inequality (i = j = k and k ≠ l), which is beyond the capability of a PDA and therefore not a CFL.
· Statement D has a complex linear equality (i = j + l), which cannot be tracked by a PDA.
· Statement E involves multiple equalities and inequalities (i = j = l and k ≠ l), making it impossible for a PDA to process and thus not context-free.