Consider the language $L=\{0^{n^2}|n0\}.$ To prove this language is not regular, a student wrote the

following proof:

Let $F=\{0^{n^2}|n0\}.$

Pick two arbitrary strings $x=0^{i^2}$ and $y=0^{j^2}$ from $L,$ where $ij.$

Then, let $z=0^{2i+1}.$

We have that $xz=0^{i^2+2i+1}=0^{(i+1{)}^2}$inL. On the other hand, $yz=0^{j^2+2i+1},$ and

$j^2+2i+1$ is not a perfect square, because we can bound it between two consecutive

perfect squares:

$yz=0^{j^2+2i+1}!$inLFLFLx,yzijzxyxzyzxz$=0^{i^2+2i+1}{0}^{i^2+2i+1}=0^{(i+1{)}^2}{0}^{(i+1{)}^2}$inLyz$=0^{j^2+2i+1}{j}^2+2j+1=(j+1{)}^{2}i>ji=ji.$

$(b)$ Without loss $of$ generality, $we$ can assume that $i>j.$

$(c)$ Without loss $of$ generality, $we$ can assume that $i=j.j^2+2i+1$

$(g)j^2+2j+1=(j+1{)}^2$

$To$ fix the proof, $we$ should add which $of$ the following sentences after the second sentence $of$ the proof?

$(a)$ Without loss $of$ generality, $we$ can assume that $i.$

$(b)$ Without loss $of$ generality, $we$ can assume that $i>j.$

$(c)$ Without loss $of$ generality, $we$ can assume that $i=j.j^2$

$(f)j^2+2i+1$

$(g)j^2+2j+1=(j+1{)}^2$

$To$ fix the proof, $we$ should add which $of$ the following sentences after the second sentence $of$ the proof?

$(a)$ Without loss $of$ generality, $we$ can assume that $i.$

$(b)$ Without loss $of$ generality, $we$ can assume that $i>j.$

$(c)$ Without loss $of$ generality, $we$ can assume that $i=j.j^2$

Thus, $yz=0^{j^2+2i+1}!$inL.

$To$ conclude, $we$ have shown that $Fis$ a fooling set for $L.$ Since $Fis$ infinite, $Lis$ not regular.

But, there's something wrong with this proof! Given the above definitions $ofx,y,$ and $z,$ give values $of$ i and

$j$ such that $z$ fails $to$ distinguish $x$ and $y(i.e.xz$ and $yz$ are either both accepted $or$ both rejected$).$

Which statement from the proof $is$ not necessarily true?

$(a)xz=0^{i^2+2i+1}$

$(b)0^{i^2+2i+1}=0^{(i+1{)}^2}$

$(c)0^{(i+1{)}^2}$inL

$(d)yz=0^{j^2+2i+1}$

$(e)j^2$

$(f)j^2+2i+1$

$(g)j^2+2j+1=(j+1{)}^2$

$To$ fix the proof, $we$ should add which $of$ the following sentences after the second sentence $of$ the proof?

$(a)$ Without loss $of$ generality, $we$ can assume that $i.$

$(b)$ Without loss $of$ generality, $we$ can assume that $i>j.$

$(c)$ Without loss $of$ generality, $we$ can assume that $i=j.$