Problem 1 - Applied Discrete Structures

Question

Let \(A_{1}=\\{1,2,3,4\\}, A_{2}=\\{4,5,6\\},\) and \(A_{3}=\\{6,7,8\\} .\) Let \(r_{1}\) be the relation from \(A_{1}\) into \(A_{2}\) defined by \(r_{1}=\\{(x, y) \mid y-x=2\\},\) and let \(r_{2}\) be the relation from \(A_{2}\) into \(A_{3}\) defined by \(r_{2}=\\{(x, y) \mid y-x=1\\}\). (a) Determine the adjacency matrices of \(r_{1}\) and \(r_{2}\). (b) Use the definition of composition to find \(r_{1} r_{2}\). (c) Verify the result in part b by finding the product of the adjacency matrices of \(r_{1}\) and \(r_{2}\)

Step-by-Step Solution

Answer

The adjacency matrices are \(M_{r_1} = \begin{pmatrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\) and \(M_{r_2} = \begin{pmatrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}\). Composition \(r_1 r_2 = \{(3, 6), (4, 7)\}\) and verified by matrix product \(M_{r_1 r_2} = \begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}\).

Step 1: Calculate Adjacency Matrix of r1

The relation \(r_1\) is defined by \(r_1 = \{(x, y) \mid y-x=2\}\) with domain \(A_1 = \{1, 2, 3, 4\}\) and codomain \(A_2 = \{4, 5, 6\}\). Evaluate possible pairs:- \((x, y) = (2, 4)\), since \(4 - 2 = 2\).- \((x, y) = (3, 5)\), since \(5 - 3 = 2\).- \((x, y) = (4, 6)\), since \(6 - 4 = 2\).The adjacency matrix \(M_{r_1}\) is a 4x3 matrix, where rows are elements of \(A_1\) and columns are elements of \(A_2\):\[M_{r_1} = \begin{pmatrix}0 & 0 & 0 \1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \\end{pmatrix}\]

Step 2: Calculate Adjacency Matrix of r2

The relation \(r_2\) is defined by \(r_2 = \{(x, y) \mid y-x=1\}\) with domain \(A_2 = \{4, 5, 6\}\) and codomain \(A_3 = \{6, 7, 8\}\). Evaluate possible pairs:- \((x, y) = (5, 6)\), since \(6 - 5 = 1\).- \((x, y) = (6, 7)\), since \(7 - 6 = 1\).The adjacency matrix \(M_{r_2}\) is a 3x3 matrix, where rows are elements of \(A_2\) and columns are elements of \(A_3\):\[M_{r_2} = \begin{pmatrix}0 & 0 & 0 \1 & 0 & 0 \0 & 1 & 0 \\end{pmatrix}\]

Step 3: Find the Composition of r1 and r2

The composition \(r_1 r_2\) is the set of pairs \((x, y)\) such that there exists a \(z\) with \((x, z) \in r_1\) and \((z, y) \in r_2\).- For \((x, z) = (3, 5)\) in \(r_1\), and \((z, y) = (5, 6)\) in \(r_2\), we find \((3, 6)\) in \(r_1 r_2\).- For \((x, z) = (4, 6)\) in \(r_1\), and \((z, y) = (6, 7)\) in \(r_2\), we find \((4, 7)\) in \(r_1 r_2\).Thus, \[ r_1 r_2 = \{(3, 6), (4, 7)\} \].

Step 4: Verify by Matrix Multiplication

Multiply the adjacency matrices \(M_{r_1}\) and \(M_{r_2}\). The resulting matrix \(M_{r_1 r_2}\) is generated by matrix multiplication:For the position of the first row:\( (M_{r_1 r_2})_{1,1} = 0*0 + 0*1 + 0*0 = 0 \)\( (M_{r_1 r_2})_{1,2} = 0*0 + 0*0 + 0*1 = 0 \)\( (M_{r_1 r_2})_{1,3} = 0*0 + 0*0 + 0*0 = 0 \)Continue for all elements:\[M_{r_1 r_2} = \begin{pmatrix}0 & 0 & 0 \0 & 0 & 0 \1 & 0 & 0 \0 & 1 & 0 \\end{pmatrix}\]Verify that non-zero entries at positions \((3, 6)\) and \((4, 7)\) match the composition set \((3, 6), (4, 7)\) derived earlier.

Key Concepts

Adjacency Matrix

In graph theory, an adjacency matrix is a means of representing the connections between nodes or elements in a set. It's a very visual way to see relationships and is utilized in various areas of discrete mathematics to model the connections. In this exercise, we dealt with two sets of data: one for relation \(r_1\) and another for \(r_2\). Each element of the matrix describes whether a direct link exists between the pairs. When we have a relation from set \(A_1\) to set \(A_2\), the adjacency matrix will have rows representing \(A_1\) and columns representing \(A_2\). Similarly, for \(r_2\), the rows represent \(A_2\) and the columns represent \(A_3\).

• A '1' in the matrix indicates a connection between two elements, which meets the defined criteria in the relation.
• A '0' shows no direct relationship.

This creates an easy-to-read visual map of relationships.

Relation Composition

The composition of two relations, noted as \(r_1 r_2\), is like connecting two maps based on a shared path. Think about it as taking a route where you first go from point \(x\) to \(z\) using \(r_1\), and then from \(z\) to \(y\) using \(r_2\). For a pair \((x, y)\) to be in the composition \(r_1 r_2\), there must be a middle point \(z\) such that \((x, z) \in r_1\) and \((z, y) \in r_2\).

• This concept is crucial for understanding how different mappings can be linked through a series of steps.
• In our problem, we identified the steps that allowed this bridging and found pairs like \((3, 6)\) and \((4, 7)\).

Understanding relation composition is essential in piecing together complex systems from simpler parts.

Matrix Multiplication

Matrix multiplication in the context of adjacency matrices helps us verify the composition of relations. When multiplying matrices \(M_{r_1}\) and \(M_{r_2}\), each element in the resulting matrix \(M_{r_1 r_2}\) is determined based on how many ways the connection between various nodes in the two matrices can be combined.

• Each entry in the result matrix is calculated by taking the sum of the products of respective elements in the corresponding row of the first matrix and column of the second matrix.
• The result will contain '1' where a valid connection exists through the relations, putting together endpoints using shared middle points.
• This mathematical operation mirrors the concept of composition by merging paths through common nodes.

Matrix multiplication offers a concrete way to calculate the same results obtained through logic and creativity in relation composition.

Discrete Mathematics

Discrete mathematics forms the backbone of logical reasoning and is crucial in computer science for problem-solving. It's the study of mathematical structures that are fundamentally discrete rather than continuous. In discrete mathematics, you deal with distinct or separate values.

• Topics include graph theory, combinatorics, algorithms, and primarily, logic. It's all about distinct paths, nodes, functions, and relations.
• The problems regarding adjacency matrices, relation composition, and matrix multiplication all fall under this blanket.
• Students leverage discrete math for modeling real-world situations, designing algorithms, and solving computational problems.

In our exercise, the discrete nature allowed us to distinctly map nodes' interconnections, calculate compositions, and confirm the results via matrix multiplication.