Answer :
Final answer:
In the poset (f, |), where 'f' is the set of all factors of 210, the maximal and greatest element is 210 and the minimal and least element is 1.
Explanation:
The student has asked to find the maximal, minimal, greatest, and least elements in the poset (f, |), where 'f' is the set of all factors of 210 and '|' denotes the divisibility relation between these factors. First, we need to find the set of factors of 210. The prime factorization of 210 is 2 × 3 × 5 × 7. From this, we can determine all the factors by creating a list of all possible products of these primes in different combinations, including 1 and the number 210 itself since every number is divisible by 1, and every number divides itself.
For the poset (f, |), the maximal element is a factor which is not divisibly by any other factor than itself. Similarly, a minimal element is a factor which does not divide any other factor than itself. Since 210 is the largest factor, it is divisibly larger than every other factor, making it the greatest element. The minimal element would be 1 since it divides every number and no other number divides it making it also the least element.
