$(15$ points$)$

$($a$)5$inA.

If xinA, then $x+4$inA.

Based on the above recursive definition, describe $A$ in at most two sentences.

$($b$)\text{'}0\text{'}$ inB.

${?}^{\text{'}}1\text{'}$ inB.

If yinB, then $\text{'}{00}^{\text{'}}$yinB and $y^{\text{'}}{11}^{\text{'}}$inB.

Write down any five non$-$empty strings in $B.$

Use at most two sentences to describe the elements of $B.$