Let B 1 2 3 6 12 18 and R be defined by xRy if and only if

Let B = {1, 2, 3, 6, 12, 18} and R be defined by xRy if and only if x|y.

a) Determine all minimal and all maximal elements of the poset.

b) Find all least and greatest elements of the poset.

Also Please show what R =, and use x and y to explaine not things like c or d.

Thanks

Solution

(a):

Minimal:

Minimal numbers are those which divide all the other numbers completely. In this set, only 1 divides all the other numbers (For example 2 does not divide 3, 3 does not divide 1 and 2 and so on). Hence 1 is the only minimal number in this set.

Maximal:

Maximal numbers are those numbers to which do not divide any other number of the set. In this set only 12 and 18 are such number which do not divide any other numbers of the set. Hence 12 and 18 are maximal elements of the given set.

(b)

Least Element:

Least element of the poset is that smallest elements which divides into all the elements of the set. Hence, 1 is the least element of the poset as well.

Greatest Element:

Greatest Element of a poset is that element which is divisible by all other elements. In this set there is no such element which can be divided by rest of the elements. So greatest element does not exist for this poset.