Methods for Representing Relations

Which of the following is a method for representing relations?

• Using Graphs

• Using Matrices

• Using Equations

• Using Tables

What is another method for representing relations besides using matrices?

• Using Graphs

• Using Digraphs

• Using Equations

• Using Tables

How can a relation between finite sets be represented?

• Using a zero-one matrix

• Using a graph

• Using a list

• Using a tree

What does the entry $ m_{ij} $ in the matrix $ M_R $ represent when $ a_i $ is related to $ b_j $?

• 1

• 0

• -1

• 2

What is the value of $ m_{ij} $ if $ (a_i, b_j) \notin R $?

• 0

• 1

• -1

• 2

Suppose that A = {1,2,3} and B = {1,2}. Let R be the relation from A to B containing (a,b) if a ∈ A, b ∈ B, and a > b. What is the matrix representing R (assuming the ordering of elements is the same as the increasing numerical order)?

• \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 1 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}\)

• \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 1 & 1 \\ 0 & 0 \\ 1 & 0 \end{bmatrix}\)

What condition must be met for a relation \( R \) to be reflexive?

• All elements on the main diagonal of \( M_R \) are equal to 1.

• All elements on the main diagonal of \( M_R \) are equal to 0.

• All elements off the main diagonal of \( M_R \) are equal to 1.

• All elements off the main diagonal of \( M_R \) are equal to 0.

When is a relation \( R \) considered symmetric?

• \( m_{ij} = 1 \) whenever \( m_{ji} = 1 \).

• \( m_{ij} = 0 \) whenever \( m_{ji} = 1 \).

• \( m_{ij} = 1 \) whenever \( m_{ji} = 0 \).

• \( m_{ij} = 0 \) whenever \( m_{ji} = 0 \).

What is the condition for a relation \( R \) to be antisymmetric?

• \( m_{ij} = 0 \) or \( m_{ji} = 0 \) when \( i \neq j \).

• \( m_{ij} = 1 \) or \( m_{ji} = 1 \) when \( i \neq j \).

• \( m_{ij} = 0 \) and \( m_{ji} = 0 \) when \( i = j \).

• \( m_{ij} = 1 \) and \( m_{ji} = 1 \) when \( i = j \).

Suppose that the relation R on a set is represented by the matrix: Is R reflexive, symmetric, and/or antisymmetric?

• Reflexive and symmetric, but not antisymmetric

• Reflexive, symmetric, and antisymmetric

• Symmetric and antisymmetric, but not reflexive

• Reflexive and antisymmetric, but not symmetric