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STEMS 2021 CS Cat B, Q3
Tags: thoery of compuatation
Let $\Sigma$ be a finite set. For $x,y \in \Sigma^{\ast}$, define \[x\preceq y\] if $x$ is a sub-string ([b]not necessarily contiguous[/b]) of $y$. For example, $ac \preceq abc$. We call a set $S\subseteq \Sigma^{\ast}$ [b][u]good[/u][/b] if $\forall x,y \in \Sigma^{\ast}$, $$ x\preceq y, \; y \in S \; \; \; \Rightarrow \; x\in S .$$ Prove or disprove: Every good set is regular.