We defined the relation $\overset{*}\Rightarrow$ with a basis $"\alpha\Rightarrow \alpha $ and an induction that says $\alpha\overset{*}\Rightarrow\beta$ and $\beta\Rightarrow\gamma$ imply $\alpha\overset{*}\Rightarrow\gamma.$ There are several other ways to define $\overset{*}\Rightarrow$ that also have the effect of saying that $"\overset{*}\Rightarrow$ is zero or more $\Rightarrow$ steps.$"$ Prove that the following are true$:$
- $\alpha\overset{*}\Rightarrow\beta$ if and only if there is a sequence of one or more strings. $\gamma_{1},\gamma_{2},.....\gamma_{n}$ Such that $\alpha=\gamma_{1},\beta=\gamma_{n},$ and for $i=1,2,...,n-1$ we have $\gamma_{i}\Rightarrow \gamma_{i+1}.$
- If $\alpha\overset{*}\Rightarrow\beta,$ and $\beta\overset{*}\Rightarrow\gamma,$ then $\alpha\overset{*}\Rightarrow\gamma.$ Hint$:$ Use induction on the number of steps in the derivation $\beta\overset{*}\Rightarrow\gamma.$
