Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole

Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in the set of lower bounds for $A$.

Exercise: Make sure you understand the definition of meet and join in a preorder $X$. Think of some simple finite preordered sets in which meets and joins do not exist.

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