Discrete Mathematics Hasse Diagrams

Understanding Discrete Mathematics and Hasse Diagrams

In the field of mathematics, discrete mathematics is a branch that deals with mathematical structures that are fundamentally distinct and separate. It focuses on objects that can only take on specific, countable values, rather than continuous values. Discrete mathematics plays a vital role in various areas, such as computer science, cryptography, and combinatorics.

One of the fundamental concepts in discrete mathematics is the Hasse diagram. Hasse diagrams are graphical representations of partially ordered sets (posets). A poset consists of a set of elements and a binary relation that defines a partial order among the elements. In other words, it describes the relationships between the elements based on a specific criterion.

Constructing a Hasse Diagram

To construct a Hasse diagram, we follow a few simple steps:

1. List all the elements of the poset.
2. Identify the relationships between the elements based on the given criterion.
3. Represent the elements as nodes and draw directed edges between them to indicate the relationships.
4. Avoid drawing redundant edges to maintain clarity.

Let’s explore a few examples to better understand how Hasse diagrams are constructed.

Example 1: Divisibility Relation

Consider the set of positive integers from 1 to 10, and the relation of divisibility. In this case, an element ‘a’ is said to be divisible by another element ‘b’ if ‘a’ is a multiple of ‘b’. Let’s construct the Hasse diagram for this relation:

Step 1: List all the elements of the poset:

Elements: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Step 2: Identify the relationships between the elements:

• 2 is divisible by 1
• 3 is divisible by 1
• 4 is divisible by 1
• 4 is divisible by 2
• 5 is divisible by 1
• 5 is divisible by 5
• 6 is divisible by 1
• 6 is divisible by 2
• 6 is divisible by 3
• 7 is divisible by 1
• 8 is divisible by 1
• 8 is divisible by 2
• 8 is divisible by 4
• 9 is divisible by 1
• 9 is divisible by 3
• 10 is divisible by 1
• 10 is divisible by 2
• 10 is divisible by 5

Step 3: Represent the elements as nodes and draw directed edges:

In this Hasse diagram, the elements are represented as nodes, and the directed edges indicate the divisibility relationships. The diagram helps us visualize the partial order among the elements based on the given criterion.

Example 2: Subsets Relation

Consider a set A = {1, 2, 3}. In this example, we will construct a Hasse diagram based on the subset relation. The subset relation indicates whether one set is a subset of another set. Let’s construct the Hasse diagram for this relation:

Step 1: List all the elements of the poset:

Elements: ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

Step 2: Identify the relationships between the elements:

• ∅ ⊆ {1}
• ∅ ⊆ {2}
• ∅ ⊆ {3}
• ∅ ⊆ {1, 2}
• ∅ ⊆ {1, 3}
• ∅ ⊆ {2, 3}
• ∅ ⊆ {1, 2, 3}
• {1} ⊆ {1, 2}
• {1} ⊆ {1, 3}
• {1} ⊆ {1, 2, 3}
• {2} ⊆ {1, 2}
• {2} ⊆ {2, 3}
• {2} ⊆ {1, 2, 3}
• {3} ⊆ {1, 3}
• {3} ⊆ {2, 3}
• {3} ⊆ {1, 2, 3}
• {1, 2} ⊆ {1, 2, 3}
• {1, 3} ⊆ {1, 2, 3}
• {2, 3} ⊆ {1, 2, 3}

Step 3: Represent the elements as nodes and draw directed edges:

In this Hasse diagram, the elements are represented as nodes, and the directed edges indicate the subset relationships. The diagram helps us visualize the partial order among the elements based on the given criterion.

Conclusion

Hasse diagrams are powerful tools in discrete mathematics for representing and visualizing the partial order relationships among elements in a poset. They provide a clear and concise way to understand the structure and hierarchy within a set of elements based on a specific criterion. By following the steps mentioned earlier, you can construct Hasse diagrams for various relations and gain insights into the underlying order and relationships.