Solution to Let S = [1,2,4,5,10,20, 25, 50, 100). Then show that 5 forms a lattice under … - Sikademy
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Archangel Macsika

Let S = [1,2,4,5,10,20, 25, 50, 100). Then show that 5 forms a lattice under divisibility. Draw the Hasse diagram also.

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Let us draw the Hasse diagram of lattices, (L_1,<) and (L_2,<) where L_1 = \{1, 2, 3, 4, 6, 12\} and L_2 = \{2, 3, 6, 12, 24\} and a < b if and only if a divides b.

Note that a Hasse diagram is a graphical rendering of a partially ordered set displayed via the cover relation of the partially ordered set with an implied upward orientation. A point is drawn for each element of the poset, and line segments are drawn between these points according to the following two rules:

1. If x<y  in the poset, then the point corresponding to x appears lower in the drawing than the point corresponding to y.

2. The line segment between the points corresponding to any two elements  x and  y of the poset is included in the drawing iff  x covers y  or  y  covers x.

In our case, x<y if and only if x|y. Therefore, the Hasse diagrams are the following:

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