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Let's first break down the problem. The question asks about a Hasse diagram and involves a set \( B = \{3, 4, 6\} \). The diagram represents a partially ordered set (poset), and we are tasked with finding:

1. All upper bounds of \( B \)
2. All lower bounds of \( B \)
3. The least upper bound of \( B \)
4. The greatest lower bound of \( B \)

### Step-by-Step Explanation:

#### 1. **All Upper Bounds of \( B \)**:
The upper bounds of a subset \( B \) in a poset are the elements in the diagram that are greater than or equal to every element of \( B \). Looking at the diagram, the elements of \( B \) are 3, 4, and 6. Any element higher in the diagram than 3, 4, or 6 (directly or indirectly) can be considered an upper bound.

- From 3, we move up to 4, 5, 6, and 8.
- From 4, we move up to 5 and 8.
- From 6, we move up to 8.

Thus, the upper bounds of \( B \) are: \( \{5, 8\} \).

#### 2. **All Lower Bounds of \( B \)**:
The lower bounds of a subset \( B \) are elements that are less than or equal to every element of \( B \).

- The elements less than or equal to 3 are 1 and 3.
- The elements less than or equal to 4 are 1, 2, and 4.
- The elements less than or equal to 6 are 1, 3, and 6.

Thus, the lower bounds of \( B \) are: \( \{1, 3\} \).

#### 3. **The Least Upper Bound of \( B \)**:
The least upper bound (also known as the *supremum*) is the smallest element that is an upper bound for all elements of \( B \). From the set of upper bounds \( \{5, 8\} \), the smallest element is 5.

Thus, the least upper bound is \( 5 \).

#### 4. **The Greatest Lower Bound of \( B \)**:
The greatest lower bound (also known as the *infimum*) is the largest element that is a lower bound for all elements of \( B \). From the set of lower bounds \( \{1, 3\} \), the largest element is 3.

Thus, the greatest lower bound is \( 3 \).

### Summary:
- (i) **All upper bounds of \( B \)**: \( \{5, 8\} \)
- (ii) **All lower bounds of \( B \)**: \( \{1, 3\} \)
- (iii) **The least upper bound of \( B \)**: \( 5 \)
- (iv) **The greatest lower bound of \( B \)**: \( 3 \)

Would you like further details or clarification on any of these points?

Here are 5 related questions to explore further:
1. What is a Hasse diagram and how is it constructed for a partially ordered set (poset)?
2. How can we determine upper and lower bounds for different subsets in a poset?
3. What is the significance of the least upper bound and greatest lower bound in a poset?
4. How does the concept of bounds relate to lattices in set theory?
5. Can every subset of a poset have both a least upper bound and greatest lower bound?

**Tip**: When analyzing Hasse diagrams, remember that only direct lines between elements are significant in determining the order; diagonal or curved connections don’t necessarily imply relationships unless explicitly shown.

Define Hasse diagram. Consider the following Hasse diagram shown in Fig. and the set B = {3, 4, 6}. Find, if they exist: (i) All upper bounds of B (ii) All lower bounds of B (iii) The least upper bound of B (iv) The greatest lower bound of B

Explore how to analyze a Hasse diagram for the set B = {3, 4, 6}. Learn to identify all upper bounds, lower bounds, the least upper bound, and the greatest lower bound in a partially ordered set (poset). Suitable for undergraduate-level students studying discrete mathematics or set theory.

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