Answer to Question #273988 in Discrete Mathematics for Jayesh

Question

Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |).

a) Find the maximal elements.

b) Find the minimal elements.

c) Is there a greatest element?

d) Is there a least element?

e) Find all upper bounds of {3, 5}.

f) Find the least upper bound of {3, 5}, if it exists.

g) Find all lower bounds of {15, 45}.

h) Find the greatest lower bound of {15, 45}, if it exists.

Answer 1

We are given:

$(\{3,5,9,15,24,45, |\})$

Let,

$P=\{3,5,9,15,24,45\}$ and $Q=\{(x,y)\space |\space x\space divides\space y\}$

We first determine the Hasse diagram. The Hasse diagram is as shown below.

a)

Maximal elements are values in the Hasse diagram that do not have any elements above them.

Therefore, maximal elements are, (24,45)

b)

Minimal elements are the values in the Hasse diagram that do not have elements below them.

Thus, minimal elements are, (3,5)

c)

Greatest element only exist if there is exactly one maximal element. Since there are two maximal elements, we do not have a greatest element.

Therefore,

Greatest element does not exist.

d)

A least element only exist if there is exactly one minimal element and is equal to that minimal element. Since there are two minimal elements, we do not have a least element.

Therefore, least element does not exist.

e)

The upper bound of a set are all elements that have a downward path to all elements in the set. Thus, the upper bound of {3,5} is,

Upper bounds=9,15,45

f)

The least upper bound of a set is the upper bound that is less than all other upper bounds. Therefore, the least upper bound of {3,5} is 9.

g)

The lower bounds of a set are all elements that have an upward path to all elements in the set. Therefore, the lower bounds of {15,45} are

lower bounds={3,5,9,15}

h)

The greatest lower bound is the lower bound that is greater than all other lower bounds. Therefore, the greatest lower bound of {15,45} is 15.