Let's say I have a set K which are a set of positive integers of the divisors of 180. I want to find the minimal and maximal elements of this set.
A maximal element means that there is no element smaller than it within the set and a minimal element means there is no other element in the set that is greater than it?
If you could please provide an example with it so I can understand how to find the minimal and maximal elements.
For a partial order $<$, an element $a$ is called "maximal" if there does not exist a $b$ with $a<b$. An element $c$ is called "minimal" if there does not exist a $d$ such that $d<c$.
I think I need to emphasize to you that it's absolutely necessary to tell us what partial order you are using. That could completely alter the solution to the question. For example you could be using the regular linear order of the natural numbers. (I guess you would not be asking a question about that, though.) You could also be using the reverse ordering of the natural numbers, or divisibility or the reverse partial order by divisibility.
I'm going to guess though that you mean the partial order of divisibility on the set of divisors of $180$, so that $a<b$ means $a\neq b$ and $a$ divides $b$.
Strictly speaking, the set of nonegative divisors of $180$ contains $1$ and $180$. Now $1<b$ for every divisor $b$ of $180$ besides $1$. Similarly, $b<180$ for every divisor $b$ besides $180$.
When there is a unique maximal element above all other elements, this element is called a "maximum" element. When there is a unique minimal element below all other elements, it is called a "minimal" element. In this case, we have both a maximum and a minimum.
Since these two solutions are rather unsurprising and a little trivial, it might be good to consider a different situation, where the set we're looking at is the divisors of 180 strictly between $1$ and $180$. I encourage you to find the maximal and minimal elements of that set now :) An example of a minimal element is $2$, an an example of a maximal element is $90$.
