(Solved): 8: Recursive Definitions and Inductive Proofs Consider the set S defined recursively as follows ...

8: Recursive Definitions and Inductive Proofs Consider the set $S$ defined recursively as follows: B1: all integers are in $\mathrm{S}$. $R1$ : If $x$ is in $S$, then $\sqrt{x}$ is in $S$. $R2$ : If $x$ and $y$ are in $S$, then $x+y,x?y$, and $xy$ are in $S$. a. Are there any elements of $S$ which are not integers? If so, give an example, and explain why it's in $\mathrm{S}$. b. Are there any elements of $S$ that are not $n$th roots of integers for any $n$ ? (Remember that $\sqrt{\sqrt{k}}=\sqrt[4]{k}$.) If so give an example, and explain why it's in S. If not, explain why not.