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Prove by induction. Every partial order on a nonempty finite set has at least one minimal element.

How can I solve that question ?

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1 Answer 1

One proof method (by induction) is on the size of your set.

The basis, a singleton. There is only one element so clearly it is minimal. Assume the claim is true for sets of size $n$, now prove for $n+1$:
Choose any element of $A$, denote it by $a$ now restrict your partial order to $A\setminus\{a\}$. By the induction hypothesis there is a minimal element there, $b$. Now look at the original order - if $a<b$ show that $a$ is minimal, otherwise $b$ is still minimal.

Either way, you found a minimal element in $A$ and your partial order.

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