Show that the grammar in Example $1\mathrm{.}14$ is unambiguous.

EXAMPLE $1\mathrm{.}14$

Consider the grammar $G_1=(\{A,S\},\{a,b\},S,P_1),$ with $P_1$ consisting

of the productions

$S$aAb$|,$

$A$aAb$|.$

Here we introduce a convenient shorthand notation in which several

production rules with the same left$-$hand sides are written on the same

line, with alternative right$-$hand sides separated by $|.$ In this notation

$S$aAb$|$ stands for the two productions $S$aAb and $S.$

This grammar is equivalent to the grammar $G$ in Example $1\mathrm{.}11.$

The equivalence is easy to prove by showing that

$L(G_1)=\{a^n{b}^n$:$n0\}.$

We leave this as an exercise.