Question

Let R be the partial order relation defined on A = {2, 3, 4, 5, 6, 8, 10, 40}, where xRy means x | y.
i.Draw the Hasse diagram for R.
ii.Find the upper and lower bounds of {4, 8}.

Answer 1

Answer on Question#37631 - <math> - <discrete mathematics="">

Let $R$ be the partial order relation defined on $A = \{2, 3, 4, 5, 6, 8, 10, 40\}$ , where $xRy$ means $x \mid y$ .

i. Draw the Hasse diagram for R.

ii. Find the upper and lower bounds of $\{4, 8\}$ .

Solution:

I

2 divides 4, 6, 8, 10 and 40.

3 divides 6.

4 divides 8 and 40.

5 divides 10 and 40.

6 doesn't divide anything.

10 divides 40;

40 doesn't divide anything.

Hasse diagram for R:

II

The upper bounds of $\{4,8\}$ are all the real numbers $\geq 8$ , i.e. in the interval [8, infinity)

Similarly the lower bounds of $\{4,8\}$ are all the numbers $\leq 4$ , i.e. in the internal $(-\mathrm{infinity},2]$

Hence, the least upper bound is 8 and the greatest lower bound is 4.