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Which of the following definitions generates the same languages as $L,$ where

$L = \{x^{n}y^{n},n \geq 1\}$

1. $E \rightarrow xEy \mid xy$
2. $xy \mid x^{+}xyy^{+}$
3. $x^{+}y^{+}$

1. $(i)$ Only
2. $(i)$ and $(ii)$ only
3. $(ii)$ and $(iii)$ only
4. $(ii)$ only

option A is correct.
option A is correct because the other option i.e. b and c giving us the choice to generate no x’s or no y’s
Option A

We need to have

1. equal number of x and y, also
2. no x can come after y occurs.

1. We can ensure equal number of x and y, also requirement 2 is followed.

2,3. We have $x^+$ and $y^+$, no way to ensure equal number of x and y.

So A is correct.

by