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Given $A = \{2, 4, 6, 8, 10, 16, 18, 24, 36, 72\}$, and given the ordered set $(A, |)$, where $|$ denotes the relationship of the divide between natural numbers.

• Draw the Hasse diagram of $(A, |)$.

• Determine all the minimal and maximal elements, and any minimum and maximum of $(A, |)$.

• Determine:

$inf_A \{16, 18\} =$

$sup_A \{4, 6\} =$

My attempt:

Minimal element: $2$

Maximal element: $72$

Minimum: $2$

Maximum: doesn't exist

I don't know how to calculate $\inf_A \{16, 18\}$ and $\sup_A \{4, 6\}$.

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1 Answer

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You missed the edges 24-72 and 4-36.

$\inf_A\{16,18\}$, if it exists, is the greatest lower bound of both 16 and 18. The lower bounds of 16 are $\{2,4,8\}$ and the lower bounds of 18 are $\{2,6\}$. 2 is the only common lower bound so it is the greatest and $\inf_A\{16,18\} = 2$.

A similar effort should show you that $\sup_A\{4,6\} = 24$.

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