Draw the Hasse diagram of the following sets under the partial order relation divides and indicate which are chains. justify your answer.

1] A = {2,4,12,24}

2] A = {1,3,5,15,30}

Solution:

1] Let A = {2,4,12,24}

Relation under divisibility is

R = { (a,b) | V $a_1$ b E A and b is div by a}

R = { (2,2) (2,4) (2,12) (2,24) (4,4) (4,12) (4,24) (12,12) (12,24) (24,24)

To draw Hasse diagram, we eliminate relativity and transitivity from the relation R

$\therefore R_1 = $ { (2,4), (4,12), (12,24)}

2] A = {1,3,5,15,30}

R = { (1,1) (1,3) (1,5) (1,15) (1,30) (3,3) (3,15) (3,30) (5,5) (5,30) (15,15) (15,30) (30,30)

$\therefore R_1 = $ { (1,3), (1,5), (3,15), (5,15),(15,30)}