Math Problem Solver

Consider a set of complex numbers $\{a+bi\}$, where $a$ and $b$ are integers. The set is partially ordered by the relation $\prec$ such that $a+bi \prec c+di$ if and only if $a<c$ or $a=c$ and $b<d$. We want to find the least upper bound of a subset $S$ of this set under this partial order. To do so, we can use an iterative approach. Starting with an initial complex number $z_0$, we iteratively apply the following rule to get the next complex number $z_{n+1}$: if $z_n$ is in $S$, then $z_{n+1}=z_n+i$, otherwise $z_{n+1}=z_n+1$. The process stops when we reach a complex number that is greater than or equal to every element in $S$. The least upper bound is then the complex conjugate of the final complex number obtained. If the subset $S$ is $\{2+2i, 3+i, 1+3i\}$ and the initial complex number is $1+i$, what is the least upper bound of $S$ under this partial order? is this prompt is correct ans answerable