Let's break down and solve the problem step by step:

i) Draw a Hasse Diagram for Set $A$ with Divisibility Relation

• Set A is given as $\{2, 3, 4, 5, 6, 30, 60\}$.
• The divisibility relation means that for any two elements $a, b \in A$, there is a directed edge from $a$ to $b$ if $a$ divides $b$ (i.e., $a \mid b$).

To construct a Hasse diagram, we will:

1. Find pairs $(a, b)$ such that $a \mid b$ and there is no element $c \in A$ such that $a \mid c \mid b$. This ensures we only draw direct relations and not transitive ones.

The divisibility relations are as follows:

• $2 \mid 4$, $2 \mid 6$, $2 \mid 30$, $2 \mid 60$
• $3 \mid 6$, $3 \mid 30$, $3 \mid 60$
• $4 \mid 60$
• $5 \mid 30$, $5 \mid 60$
• $6 \mid 30$, $6 \mid 60$
• $30 \mid 60$

The Hasse diagram will represent these relations without showing transitive edges. I will generate the Hasse diagram.

ii) Find the Greatest and Least Elements

• Greatest element: The greatest element in terms of divisibility is the element that is divisible by all other elements in the set. In this case, it is 60 because all other elements in the set divide 60.

• Least element: The least element is the one that divides all other elements, which does not exist in this case because no single element divides all others.

iii) Find Lower Bound (LB), Upper Bound (UB), Least Upper Bound (LUB), and Greatest Lower Bound (GLB) for $\{6, 30\}$

For the subset $\{6, 30\}$:

• Lower Bound (LB): The LB is any element in the set that divides both 6 and 30. The divisors of both 6 and 30 are $\{1, 2, 3, 6\}$, so the lower bounds are $\{2, 3, 6\}$.

• Upper Bound (UB): The UB is any element in the set that is divisible by both 6 and 30. In this case, the upper bounds are $\{30, 60\}$.

• Least Upper Bound (LUB): The LUB is the smallest element in the set that is divisible by both 6 and 30. This is $30$.

• Greatest Lower Bound (GLB): The GLB is the largest element in the set that divides both 6 and 30. This is $6$.

Let me know if you'd like further details, or if I should draw the Hasse diagram for you.

Here are 5 related questions to explore:

1. How do we find upper bounds for a set with more than two elements?
2. Can a set have more than one greatest element?
3. What happens if a set has no least element?
4. How do we use Hasse diagrams to visualize partially ordered sets?
5. How does divisibility work in sets that include prime numbers?

Tip: When constructing a Hasse diagram, always remove transitive relations to simplify the diagram.