# Different Number of a's and b's in Context-Free Grammar

## What is a context-free grammar for a language with a different number of a's than b's?

How can we construct context-free grammar rules to generate strings with varying numbers of 'a's and 'b's?

## Answer:

A context-free grammar for the language L = {a^n b^m | n ≠ m, n, m > 0} can be defined as follows:

A context-free grammar is a set of rules used to generate strings in a formal language. In this case, we want to create a grammar where the number of 'a's is not equal to the number of 'b's, with both 'a' and 'b' occurring at least once.

The context-free grammar for this language can be defined as follows:

**S -> aSb | bSa | aTb | bTa**

**T -> aT | bT | ε**

The start symbol is S, which represents the entire language. The production rules allow for the generation of strings with a different number of 'a's and 'b's.

The rule S -> aSb generates strings with additional 'a' before and 'b' after the non-terminal symbol S, ensuring that the number of 'a's is greater than the number of 'b's.

Similarly, the rule S -> bSa generates strings with additional 'b' before and 'a' after the non-terminal symbol S, ensuring that the number of 'b's is greater than the number of 'a's.

The rule S -> aTb allows for strings where 'a' and 'b' are separated by a sequence generated by the non-terminal T.

The rule S -> bTa allows for strings where 'b' and 'a' are separated by a sequence generated by the non-terminal T. The non-terminal T allows for the generation of any number of 'a's or 'b's, including the empty string (ε).