To determine the maximal and minimal elements of the set $\{2, 4, 5, 8, 10, 12, 20, 25\}$ under the "divides" relation (denoted $|$), we analyze each element based on divisibility:

Understanding the Divisibility Relation

In this context, an element $a$ is considered "less than or equal to" another element $b$ (i.e., $a \leq b$) if $a$ divides $b$ (written $a | b$). Therefore:

• Minimal elements: Elements that are not divisible by any other element in the set, except by themselves.
• Maximal elements: Elements that do not divide any other element in the set, except themselves.

Step-by-Step Analysis

1. Identify Minimal Elements: These elements should have no divisors within the set other than themselves.

   • $2$ divides $4, 8, 10, 12, 20$.
   • $4$ divides $8, 12, 20$.
   • $5$ divides $10, 20, 25$.
   • $8$ divides $20$.
   • $10$ divides $20$.
   • $12$ divides no other elements.
   • $20$ divides no other elements.
   • $25$ divides no other elements.

   Based on this, the minimal elements are those that are not divisible by any smaller element in the set:

   • Minimal Elements: $2$ and $5$, as these are the smallest divisors.

2. Identify Maximal Elements: These elements should not divide any other element in the set except themselves.

   • $12$, $20$, and $25$ do not divide any other elements in the set.

   Therefore, the maximal elements are:

   • Maximal Elements: $12$, $20$, and $25$.

Summary

• Minimal Elements: $2$, $5$
• Maximal Elements: $12$, $20$, $25$

Would you like further details on the reasoning? Here are some related questions:

1. What are some other ways to define maximal and minimal elements in ordered sets?
2. How would adding or removing an element affect the maximal and minimal elements?
3. Can there be multiple minimal elements in every set with the divides relation?
4. How does the divides relation differ from the standard numerical comparison?
5. What are other examples of partial order relations in mathematics?

Tip: In sets with the divides relation, minimal elements are typically prime factors of other elements when possible, while maximal elements are often the largest numbers that don’t divide others.