Question $3$

I. What is the transitive closure of $\{(x,y)inZZ|y=x2\}?$

What is the reflexive transitive closure of $\{(x,y)inZZ|y=x+2vvy=x-2\}?$

$($Continued on next page$)$

Programming Languages $2023($final exam.$)$

$e(exp)$::$=n|e|e_{1}o+e_2$

$\frac{e_1{e}_1^{\text{'}}}{e_{1}o+e_2{e}_1^{\text{'}}o+e_2},\frac{e_2{e}_2^{\text{'}}}{no+e_{2}no+e_2^{\text{'}}},\frac{mis\text{t}\text{h}\text{e}\text{s}\text{u}\text{m}of{n}_1\text{a}\text{n}\text{d}{n}_2}{n_{1}o+n_{2}m}$

$\frac{e{e}^{\text{'}}}{ee},\frac{mis\text{t}\text{h}\text{e}\text{n}\text{e}\text{g}\text{a}\text{t}\text{i}\text{o}\text{n}ofn}{nm}$

$m$ is the negation of $n$

$nm$

Figure I: Syntax and operational semantics of is a meta$-$variable

for integers$)$