Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

How can I solve that question ?

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.