Problem 1 - Context Free Grammars

Give a context free grammar that produces the following set:

1.

All palindromes (a sentence that is the same when read from beginning and read from the end) on the alphabet $\{a, b \}$.

2.

All strings on the alphabet $\{a, b \}$ such that $a$ appears twice as many times as $b$.

1.

A context-free grammar generating the set of palindromes would have the productions:

$$S \rightarrow aSa \; \vert \; bSb \; \vert \; a \; \vert \; b \; \vert \; \epsilon$$

2.

A context-free grammar generating the set of all strings on the alphabet $\{a, b \}$ such that $a$ appears twice as many times as $b$ would have the productions:

$$S \rightarrow aSaSb \; \vert \; aSbSa \; \vert \; bSaSa \; \vert \; \epsilon$$