need help with the following question:
let $R$ be a binary relation.
We will define a series of relations $R_1,R_2,R_3\dots$ so that:
$R_0 = R$
$R_{i+1} = R_i\cup\{\langle x,y\rangle :\exists y(xR_iy\wedge yR_iz)\}$
we need to prove that $\bigcup_{i=0}^{\infty}R_i$ is transitive thank you:)
Denote $S=\bigcup_{i=1}^{\infty}R_i$.
First observe that $i\leq j\implies R_i\subseteq R_j\subseteq S$.
You can formally prove that by induction.
Let $xSy$ and $ySz$.
Then $xR_iy$ and $yR_jz$ for nonnegative integers $i,j$.
Then if $k=\max(i,j)$ we have $xR_ky\wedge yR_kz$ hence $xR_{k+1}z$.
Then $xSz$ because $R_{k+1}\subseteq S$.