Answer to Question #117138 in Combinatorics | Number Theory for Priya

Question

For the poset ({2,4,6,9,12,18,27,36,48,60,72},|) having the `divides’ relation, find
(a) the maximal elements
(b) the minimal elements
(c) All upper bounds of {2,9}
(d) The least upper bound of {2,9} if it exists.

Answer 1

Let the given set be $S=\{2,4,6,9,12,18,27,36,48,60,72\}$

1) We known that ,An element $m\in S$ is called the maximal element of $S$ if $m | x \implies m=x$ Where $x\in S$ .

Thus $27,48,60$ and $70$ are the maximal element.

2) We known that ,an element $a\in S$ is called the minimal element of $S$ if $x|a\implies x=a$ Where $x\in S$ .

Thus , $2,9$ are maximal element.

3)The upper bound of $\{ 2,9\}$ are 18,36,72

4) Since 18 is the least element of $\{ 18,36,72\}$ .

Thus , least upper bound of $\{2,9\}$ is 18.